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![Preface
This book has been written to be of use to mathematicians working in algebraic
(or more precisely, spectral) graph theory. It also contains material that may
be of interest to graduate students dealing with the same subject area. It is
primarily a theoretical book with an indication of possible applications, and so
it can be used by computer scientists, chemists, physicists, biologists, electrical
engineers, and other scientists who are using the theory of graph spectra in their
work.
The rapid development of the theory of graph spectra has caused the ap-
pearance of various inequalities involving spectral invariants of a graph. The
main purpose of this book is to expose those results along with their proofs,
discussions, comparisons, examples, and exercises. We also indicate some con-
jectures and open problems that might provide initiatives for further research.
The book is written to be as self-contained as possible, but we assume famil-
iarity with linear algebra, graph theory, and particularly with the basic concepts
of the theory of graph spectra. For those who need some additional material,
we recommend the books [58, 98, 102, 170].
The graphs considered here are finite, simple (so without loops or multiple
edges), and undirected, and the spectra considered in the largest part of the
book are those of the adjacency matrix, Laplacian matrix, and signless Lapla-
cian matrix of a graph. Although the results may be exposed in different ways,
say from simple to more complicated, or in parts by following their histori-
cal appearance, here we follow the concept of from general to specific, that
is, whenever possible, we give a general result, idea or method, and then its
consequences or particular cases. This concept is applied in many places, see
for example Theorem 2.2 and its consequences, the whole of Subsection 2.1.2
or Theorem 2.19 and its consequences.
ix](https://image.slidesharecdn.com/100481-250531074453-55eaa1b1/75/Inequalities-for-Graph-Eigenvalues-1st-Edition-Zoran-Stanic-16-2048.jpg)
![x Preface
We briefly outline the content of the book. In Chapter 1 we fix the termi-
nology and notation, introduce the matrices associated with a graph, give the
necessary results, select possible applications, and give more details about the
content. In this respect, the last section of this chapter can be considered as
an extension of this Preface. In Chapters 2–4 we consider inequalities that
include the largest, the least, and the second largest eigenvalue of the adja-
cency matrix of a graph, respectively. The last section of Chapter 4 contains
the lists of graphs obtained, together with some additional data. The remain-
ing, less investigated, eigenvalues of the adjacency matrix are considered in
Chapter 5. Chapters 6 and 7 deal with the inequalities for single eigenvalues
of the Laplacian and signless Laplacian matrix. The inequalities that include
multiple eigenvalues of any of three spectra considered before are singled out
in Chapter 8. In Chapter 9 we consider the normalized Laplacian matrix, the
Seidel matrix, and the distance matrix of a graph.
Each of Chapters 2–9 contains theoretical results, comments (including ad-
ditional explanations, similar results or possible applications), comparisons of
inequalities obtained, and numerical or other examples. Each of these chapters
ends with exercises and notes. The exercises contain selected problems or a
small number of the previous results whose proofs were omitted. The notes
contain brief surveys of unmentioned results and directions to the correspond-
ing literature.
Spectral inequalities occupy a central place in this book. Mostly, they are
lower or upper bounds for selected eigenvalues. Apart from these, we con-
sider some results written rather in the form of an inequality that bounds some
structural invariant in terms of graph eigenvalues (and possibly some other
quantities) or, as we have already said, inequalities that include more than one
eigenvalue. All inequalities exposed are listed at the end of the book.
In an informal sense, extremal graph theory deals with the problem of de-
termining extremal graphs for a given graph invariant in a set of graphs with
prescribed properties. In the context of the theory of graph spectra, the invari-
ant in question is a fixed eigenvalue of a matrix associated with a graph or a
spectral invariant based on a number of graph eigenvalues (like the graph en-
ergy). Extremal graphs for a given spectral invariant in various sets of graphs
are widely considered.
The terminology and notation are mainly taken from [98, 102], and they can
also be found in similar literature. However, since there is some overlap in the
wider notation used, we have made some small adjustments for this book only.
The author is grateful to Dragoš Cvetković and Vladimir Nikiforov, who](https://image.slidesharecdn.com/100481-250531074453-55eaa1b1/75/Inequalities-for-Graph-Eigenvalues-1st-Edition-Zoran-Stanic-17-2048.jpg)
![2 Introduction
A graph H obtained from a given graph G by deleting some vertices (to-
gether with their edges incident) is called an induced subgraph of G. In this
case, we also say that H is induced in G, and that G is a supergraph of H. We
say that a graph G is H-free if it does not contain H as an induced subgraph. A
subgraph of G is any graph H satisfying V(H) ⊆ V(G) and E(H) ⊆ E(G). If
V(H) = V(G), H is called a spanning subgraph of G.
If U ⊂ V(G), then we write G[U] to denote the induced subgraph of G with
vertex set U and two vertices being adjacent if and only if they are adjacent in
G. Similarly, an induced subgraph of G obtained by deleting a set of vertices
V
⊆ V(G) is denoted by G−V
(rather than G[V(G)V
]). If V
consists of a
single vertex v, we simply write G−v (instead of G− {v}). Similarly, G−E
and G − e designate the deletion of a subset of edges E and a single edge e,
respectively. By G+e we denote a graph obtained from G by inserting a single
edge. If V1 and V2 are disjoint subsets of V(G), then m(V1) and m(V1,V2) stand
for the number of edges in G[V1] and the number of edges with one end in V1
and the other in V2, respectively.
The degree du of a vertex u (in a graph G) is the number of edges incident
with it. In particular, the minimal and the maximal vertex degrees are denoted
by δ and Δ, respectively. We say that a graph G is regular of degree r (or rG)
if all its vertices have degree r. If so, then we usually say that G is r-regular.
The complete graph on n vertices, Kn, is a graph whose every pair of vertices
is joined by an edge. A regular graph of degree 3 is called a cubic graph. The
unique (2n−2)-regular graph on 2n (n ≥ 1) vertices is called a cocktail party,
and is denoted by CP(n). Obviously, it is an (n−1)-regular graph. A bidegreed
graph has exactly two distinct vertex degrees. The edge degree of an edge uv
is defined as du + dv − 2 (i.e., it is the number of edges that have a common
vertex with uv).
The set of neighbours (or the open neighbourhood) of a vertex u is denoted
by N(u). The closed neighbourhood of u is denoted by N[u] (= {u} ∪ N(u)).
The average degree of vertices in N(u) is denoted by mu, and it is also called
the average 2-degree of u.
A graph is said to be properly coloured if each vertex is coloured so that
adjacent vertices have different colours. G is k-colourable if it can be properly
coloured by k colours. The chromatic number χ is k if G is k-colourable and
not (k − 1)-colourable. G is called bipartite if its chromatic number is 1 or 2.
The vertex set of a bipartite graph can be partitioned into two parts (or colour
classes) X and Y such that every edge of G joins a vertex in X with a vertex in
Y. A graph is called complete bipartite if every vertex in one part is adjacent to
every vertex in the other part. If |X| = n1 and |Y| = n2, the complete bipartite](https://image.slidesharecdn.com/100481-250531074453-55eaa1b1/75/Inequalities-for-Graph-Eigenvalues-1st-Edition-Zoran-Stanic-21-2048.jpg)
![1.1 Graph-theoretic notions 5
1.1.1 Some graphs
The vertex with maximal degree in the star K1,n is called the centre of the
star. The double star DS(n1,n2) is a graph obtained from the stars K1,n1−1 and
K1,n2−1 by inserting an edge between their centres.
A starlike tree Si1,i2,...,ik
is a tree with exactly one vertex of degree greater
than two such that the removal of this vertex gives rise to paths Pi1, Pi2 ,...,Pik
.
For k = 3, the corresponding starlike tree is often called a T-shape tree.
A caterpillar is a tree in which the removal of all pendant vertices gives
a path. Let the vertices of a path Pk (k ≥ 3) be labelled 1,2,...,k (in natural
order), then T(m2,m3,...,mk−1) denotes the caterpillar obtained by attaching
mi pendant vertices at the ith vertex of Pk (2 ≤ i ≤ k − 1). If a caterpillar is
obtained by attaching just a few pendant vertices at the same path, we use the
shorter notation T
i1,i2,...,il
n , where n denotes the order and ij (1 ≤ j ≤ l) indicates
attaching a pendant vertex at the vertex labelled ij (2 ≤ ij ≤ k − 1). A closed
caterpillar is a unicyclic graph in which the removal of all pendant vertices
gives a cycle.
An open quipu is a tree with maximal vertex degree 3 such that all vertices
of degree 3 lie on a path. A closed quipu is a unicyclic graph with maximal
vertex degree 3 such that all vertices of degree 3 lie on a cycle.
A cactus is a connected graph G such that any two cycles induced in G have
at most one common vertex.
The comet C(k,l) is a tree obtained by attaching k pendant vertices at one
end of the path Pl. The double comet DC(k,l) is a tree obtained by attaching k
pendant vertices at one end of the path Pl and another k pendant vertices at the
other end of the same path.
The kite K(k,l) is a graph obtained by identifying one end of the path Pl+1
with a vertex of the complete graph Kk. The double kite DK(k,l) is a graph
obtained by identifying one end of the path Pl+2 with a vertex of the complete
graph Kk and the other end of the same path with a vertex of another complete
graph Kk.
The pineapple P(k,l) is a graph obtained by attaching l pendant vertices at
a vertex of Kk.
C(k,l), K(k,l), and P(k,l) have k + l vertices, while DC(k,l) and DK(k,l)
have 2k + l vertices.
Let GD = GD(n1,n2,...,nD+1) denote the graph defined as follows:V(GD
) =
D+1
i=1
Vi, where GD[Vi] ∼
= Kni (1 ≤ i ≤ D+1) and](https://image.slidesharecdn.com/100481-250531074453-55eaa1b1/75/Inequalities-for-Graph-Eigenvalues-1st-Edition-Zoran-Stanic-24-2048.jpg)
![6 Introduction
GD
[Vi ∪Vj] ∼
=
Kni+nj , if |i− j| = 1,
Kni ∪Knj , otherwise.
The graph GD consists of a chain of D + 1 cliques Kn1 ,Kn2 ,...,KnD+1 , where
neighbouring cliques are fully interconnected (i.e., each vertex in one is ad-
jacent to all vertices in the other). According to this, we name this graph the
clique chain graph. Its order is n = ∑D+1
i=1 ni.
Similarly, let GD
∗ = GD
∗ (n1,n2,...,nD+1) denote the graph of the same order
defined as follows: V(GD
∗ ) =
D+1
i=1
Vi, where GD
∗ [Vi] ∼
= niK1 (1 ≤ i ≤ D+1) and
GD
∗ [Vi ∪Vj] ∼
=
Kni,nj , if |i− j| = 1,
(ni +nj)K1, otherwise.
This graph consists of a chain of D + 1 co-cliques n1K1,n2K1,...,nD+1K1,
where neighbouring co-cliques are fully interconnected. We name it the co-
clique chain graph. Observe that GD
∗ is bipartite.
Recall that a multigraph includes the possible existence of multiple edges
between any two vertices or loops (i.e. edges with both endvertices identical).
We say that a petal is added to a graph when we add a pendant vertex and then
duplicate the edge incident with it.
The line graph2 Line(G) of a multigraph G is the graph whose vertices are
the edges of G, with two vertices adjacent whenever the corresponding edges
have exactly one common vertex.
Let G be a graph with vertex set V = {v1,v2,...,vn}, and let a1,a2,...,an
be non-negative integers. The generalized line graph Line(G;a1,a2,...,an) is
the graph Line(
G), where
G is the multigraph G(a1,a2,...,an) obtained from
G by adding ai petals at vertex vi (1 ≤ i ≤ n).
We introduce two classes of graphs called nested split graphs and double
nested graphs. For these two classes of graphs we use the common name nested
graphs.
A nested split graph (NSG for short) is a graph which does not contain any
of the graphs P4,C4 or 2K2 as an induced subgraph. This name is derived from
its structure; it is also called a threshold graph (for more details, see [409]).
We describe the structure of connected NSGs. The vertex set of any such graph
consists of a co-clique and a clique, where both the co-clique and the clique
are partitioned into h cells U1,U2,...,Uh and V1,V2,...,Vh, respectively. Then
2 The line graph is often denoted by L(G), but in this book L(G) is reserved for the Laplacian
matrix (see Section 1.2).](https://image.slidesharecdn.com/100481-250531074453-55eaa1b1/75/Inequalities-for-Graph-Eigenvalues-1st-Edition-Zoran-Stanic-25-2048.jpg)
Inequalities for Graph Eigenvalues 1st Edition Zoran Stanić
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- 8. LONDON MATHEMATICAL SOCIETYLECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at http://www.cambridge.org/mathematics 298 Higher operads, higher categories, T. LEINSTER (ed) 299 Kleinian groups and hyperbolic 3-manifolds, Y. KOMORI, V. MARKOVIC & C. SERIES (eds) 300 Introduction to Möbius differential geometry, U. HERTRICH-JEROMIN 301 Stable modules and the D(2)-problem, F.E.A. JOHNSON 302 Discrete and continuous nonlinear Schrödinger systems, M.J. ABLOWITZ, B. PRINARI & A.D. TRUBATCH 303 Number theory and algebraic geometry, M. REID & A. SKOROBOGATOV (eds) 304 Groups St Andrews 2001 in Oxford I, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 305 Groups St Andrews 2001 in Oxford II, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 306 Geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds) 307 Surveys in combinatorics 2003, C.D. WENSLEY (ed.) 308 Topology, geometry and quantum field theory, U.L. TILLMANN (ed) 309 Corings and comodules, T. BRZEZINSKI & R. WISBAUER 310 Topics in dynamics and ergodic theory, S. BEZUGLYI & S. KOLYADA (eds) 311 Groups: topological, combinatorial and arithmetic aspects, T.W. MÜLLER (ed) 312 Foundations of computational mathematics, Minneapolis 2002, F. CUCKER et al (eds) 313 Transcendental aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) 314 Spectral generalizations of line graphs, D. CVETKOVIĆ, P. ROWLINSON & S. SIMIĆ 315 Structured ring spectra, A. BAKER & B. RICHTER (eds) 316 Linear logic in computer science, T. EHRHARD, P. RUET, J.-Y. GIRARD & P. SCOTT (eds) 317 Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI & N.P. SMART (eds) 318 Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY 319 Double affine Hecke algebras, I. CHEREDNIK 320 L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁŘ (eds) 321 Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) 323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) 324 Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) 325 Lectures on the Ricci flow, P. TOPPING 326 Modular representations of finite groups of Lie type, J.E. HUMPHREYS 327 Surveys in combinatorics 2005, B.S. WEBB (ed) 328 Fundamentals of hyperbolic manifolds, R. CANARY, D. EPSTEIN & A. MARDEN (eds) 329 Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) 330 Noncommutative localization in algebra and topology, A. RANICKI (ed) 331 Foundations of computational mathematics, Santander 2005, L.M PARDO, A. PINKUS, E. SÜLI & M.J. TODD (eds) 332 Handbook of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) 333 Synthetic differential geometry (2nd Edition), A. KOCK 334 The Navier–Stokes equations, N. RILEY & P. DRAZIN 335 Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER 336 Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE 337 Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics 2007, A. HILTON & J. TALBOT (eds) 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) 348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) 349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH 352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds) 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) 359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds)
- 9. 360 Zariski geometries,B. ZILBER 361 Words: Notes on verbal width in groups, D. SEGAL 362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA 363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) 364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) 365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) 367 Random matrices: High dimensional phenomena, G. BLOWER 368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) 369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ 370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH 371 Conformal fractals, F. PRZYTYCKI & M. URBAŃSKI 372 Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) 373 Smoothness, regularity and complete intersection, J. MAJADAS & A. G. RODICIO 374 Geometric analysis of hyperbolic differential equations: An introduction, S. ALINHAC 375 Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds) 376 Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) 377 An introduction to Galois cohomology and its applications, G. BERHUY 378 Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds) 379 Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) 380 Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T.C WALL (eds) 381 Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTER- NITZ (eds) 382 Forcing with random variables and proof complexity, J. KRAJÍČEK 383 Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 384 Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) 386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER 387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds) 388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds) 389 Random fields on the sphere, D. MARINUCCI & G. PECCATI 390 Localization in periodic potentials, D.E. PELINOVSKY 391 Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER 392 Surveys in combinatorics 2011, R. CHAPMAN (ed) 393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds) 394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds) 395 How groups grow, A. MANN 396 Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA 397 Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds) 398 Mathematical models in contact mechanics, M. SOFONEA & A. MATEI 399 Circuit double cover of graphs, C.-Q. ZHANG 400 Dense sphere packings: a blueprint for formal proofs, T. HALES 401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU 402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J.L. RODRIGO & W. SADOWSKI (eds) 403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS & A. SZANTO (eds) 404 Operator methods for boundary value problems, S. HASSI, H.S.V. DE SNOO & F.H. SZAFRANIEC (eds) 405 Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) 406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) 407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL 408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds) 409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds) 410 Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI 411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCÍA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds) 412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS 413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds) 414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT 417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTAŢĂ & M. POPA (eds) 418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) 419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM 420 Arithmetic and geometry, L. DIEULEFAIT et al (eds) 421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) 422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds) 423 Inequalities for Graph Eigenvalues, Z. STANIĆ 424 Surveys in Combinatorics 2015, A. CZUMAJ et al (eds)
- 10. London Mathematical SocietyLecture Note Series: 423 Inequalities for Graph Eigenvalues ZORAN STANIĆ University of Belgrade, Serbia
- 11. University Printing House,Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107545977 © Zoran Stanić 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Stanić, Zoran, 1975– Inequalities for graph eigenvalues / Zoran Stanic, Univerzitetu Beogradu, Serbia. pages cm. – (London Mathematical Society lecture note series ; 423) Includes bibliographical references and index. ISBN 978-1-107-54597-7 (Paper back : alk. paper) 1. Graph theory. 2. Eigenvalues I. Title. QA166.S73 2015 512.9436–dc23 2015011588 ISBN 978-1-107-54597-7 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 12. Contents Preface page ix 1Introduction 1 1.1 Graph-theoretic notions 1 1.1.1 Some graphs 5 1.2 Spectra of graphs 8 1.2.1 Spectrum of a graph 10 1.2.2 Laplacian spectrum of a graph 12 1.2.3 Signless Laplacian spectrum of a graph 14 1.2.4 Relations between A, L, and Q 16 1.3 Some more specific elements of the theory of graph spectra 18 1.3.1 Eigenvalue interlacing 18 1.3.2 Small perturbations 19 1.3.3 Hoffman program 20 1.3.4 Star complement technique 21 1.4 A few more words 22 1.4.1 Selected applications 22 1.4.2 Spectral inequalities and extremal graph theory 26 1.4.3 Computer help 27 2 Spectral radius 28 2.1 General inequalities 28 2.1.1 Walks in graphs 29 2.1.2 Graph diameter 38 2.1.3 Other inequalities 43 2.2 Inequalities for spectral radius of particular types of graph 50 2.2.1 Bipartite graphs 51 2.2.2 Forbidden induced subgraphs 55 v
- 13. vi Contents 2.2.3 Nearlyregular graphs 56 2.2.4 Nested graphs 58 2.3 Extremal graphs 63 2.3.1 Graphs whose spectral radius does not exceed 3 √ 2 2 63 2.3.2 Order, size, and maximal spectral radius 66 2.3.3 Diameter and extremal spectral radius 68 2.3.4 Trees 71 2.3.5 Various results 74 2.3.6 Ordering graphs 76 Exercises 82 Notes 85 3 Least eigenvalue 87 3.1 Inequalities 87 3.1.1 Bounds in terms of order and size 88 3.1.2 Inequalities in terms of clique number, inde- pendence number or chromatic number 90 3.2 Graphs whose least eigenvalue is at least −2 92 3.3 Graphs with minimal least eigenvalue 95 3.3.1 Least eigenvalue under small graph perturbations 96 3.3.2 Graphs of fixed order and size 98 3.3.3 Graphs with prescribed properties 100 Exercises 102 Notes 103 4 Second largest eigenvalue 105 4.1 Inequalities 105 4.1.1 Regular graphs 106 4.1.2 Trees 113 4.2 Graphs with small second largest eigenvalue 115 4.2.1 Graphs with λ2 ≤ 1 3 or λ2 ≤ √ 2−1 115 4.2.2 The golden section bound 117 4.2.3 Graphs whose second largest eigenvalue does not exceed 1 118 4.2.4 Trees with λ2 ≤ √ 2 123 4.2.5 Notes on reflexive cacti 125 4.2.6 Regular graphs 126 4.3 Appendix 134 Exercises 143 Notes 144
- 14. Contents vii 5 Othereigenvalues of the adjacency matrix 146 5.1 Bounds for λi 146 5.2 Graphs with λ3 0 149 5.3 Graphs G with λn−1(G) and λn−1(G) ≥ −1 150 Exercises 153 Notes 154 6 Laplacian eigenvalues 155 6.1 General inequalities for L-spectral radius 155 6.1.1 Upper bounds 156 6.1.2 Lower bounds 161 6.2 Bounding L-spectral radius of particular types of graph 163 6.2.1 Triangle-free graphs 163 6.2.2 Triangulation graphs 165 6.2.3 Bipartite graphs and trees 167 6.3 Graphs with small L-spectral radius 168 6.4 Graphs with maximal L-spectral radius 169 6.4.1 Graphs with μ1 = n 169 6.4.2 Various graphs 171 6.5 Ordering graphs by L-spectral radius 174 6.6 General inequalities for algebraic connectivity 176 6.6.1 Upper and lower bounds 179 6.6.2 Bounding graph invariants by algebraic con- nectivity 184 6.6.3 Isoperimetric problem and graph expansion 185 6.7 Notes on algebraic connectivity of trees 188 6.8 Graphs with extremal algebraic connectivity 190 6.9 Ordering graphs by algebraic connectivity 193 6.10 Other L-eigenvalues 193 6.10.1 Bounds for μi 194 6.10.2 Graphs with small μ2 or μ3 197 Exercises 198 Notes 202 7 Signless Laplacian eigenvalues 204 7.1 General inequalities for Q-spectral radius 204 7.1.1 Transferring upper bounds for μ1 205 7.2 Bounds for Q-spectral radius of connected nested graphs 210 7.3 Graphs with small Q-spectral radius 213 7.4 Graphs with maximal Q-spectral radius 214 7.4.1 Order, size, and maximal Q-spectral radius 214
- 15. viii Contents 7.4.2 Otherresults 216 7.5 Ordering graphs by Q-spectral radius 217 7.6 Least Q-eigenvalue 217 7.6.1 Upper and lower bounds 218 7.6.2 Small graph perturbations and graphs with extremal least Q-eigenvalue 221 7.7 Other Q-eigenvalues 222 Exercises 227 Notes 229 8 Inequalities for multiple eigenvalues 231 8.1 Spectral spread 231 8.1.1 Upper and lower bounds 231 8.1.2 Q-Spread and L-spread 234 8.1.3 Extremal graphs 234 8.2 Spectral gap 236 8.3 Inequalities of Nordhaus–Gaddum type 238 8.4 Other inequalities that include two eigenvalues 240 8.5 Graph energy 243 8.6 Estrada index 245 Exercises 247 Notes 250 9 Other spectra of graphs 251 9.1 Normalized L-eigenvalues 251 9.1.1 Upper and lower bounds for μ1 and μn−1 253 9.2 Seidel matrix eigenvalues 255 9.3 Distance matrix eigenvalues 255 9.3.1 Upper and lower bounds for δ1 257 9.3.2 Graphs with small δ2 or large δn 260 Exercises 261 Notes 262 References 265 Inequalities 290 Index 294
- 16. Preface This book hasbeen written to be of use to mathematicians working in algebraic (or more precisely, spectral) graph theory. It also contains material that may be of interest to graduate students dealing with the same subject area. It is primarily a theoretical book with an indication of possible applications, and so it can be used by computer scientists, chemists, physicists, biologists, electrical engineers, and other scientists who are using the theory of graph spectra in their work. The rapid development of the theory of graph spectra has caused the ap- pearance of various inequalities involving spectral invariants of a graph. The main purpose of this book is to expose those results along with their proofs, discussions, comparisons, examples, and exercises. We also indicate some con- jectures and open problems that might provide initiatives for further research. The book is written to be as self-contained as possible, but we assume famil- iarity with linear algebra, graph theory, and particularly with the basic concepts of the theory of graph spectra. For those who need some additional material, we recommend the books [58, 98, 102, 170]. The graphs considered here are finite, simple (so without loops or multiple edges), and undirected, and the spectra considered in the largest part of the book are those of the adjacency matrix, Laplacian matrix, and signless Lapla- cian matrix of a graph. Although the results may be exposed in different ways, say from simple to more complicated, or in parts by following their histori- cal appearance, here we follow the concept of from general to specific, that is, whenever possible, we give a general result, idea or method, and then its consequences or particular cases. This concept is applied in many places, see for example Theorem 2.2 and its consequences, the whole of Subsection 2.1.2 or Theorem 2.19 and its consequences. ix
- 17. x Preface We brieflyoutline the content of the book. In Chapter 1 we fix the termi- nology and notation, introduce the matrices associated with a graph, give the necessary results, select possible applications, and give more details about the content. In this respect, the last section of this chapter can be considered as an extension of this Preface. In Chapters 2–4 we consider inequalities that include the largest, the least, and the second largest eigenvalue of the adja- cency matrix of a graph, respectively. The last section of Chapter 4 contains the lists of graphs obtained, together with some additional data. The remain- ing, less investigated, eigenvalues of the adjacency matrix are considered in Chapter 5. Chapters 6 and 7 deal with the inequalities for single eigenvalues of the Laplacian and signless Laplacian matrix. The inequalities that include multiple eigenvalues of any of three spectra considered before are singled out in Chapter 8. In Chapter 9 we consider the normalized Laplacian matrix, the Seidel matrix, and the distance matrix of a graph. Each of Chapters 2–9 contains theoretical results, comments (including ad- ditional explanations, similar results or possible applications), comparisons of inequalities obtained, and numerical or other examples. Each of these chapters ends with exercises and notes. The exercises contain selected problems or a small number of the previous results whose proofs were omitted. The notes contain brief surveys of unmentioned results and directions to the correspond- ing literature. Spectral inequalities occupy a central place in this book. Mostly, they are lower or upper bounds for selected eigenvalues. Apart from these, we con- sider some results written rather in the form of an inequality that bounds some structural invariant in terms of graph eigenvalues (and possibly some other quantities) or, as we have already said, inequalities that include more than one eigenvalue. All inequalities exposed are listed at the end of the book. In an informal sense, extremal graph theory deals with the problem of de- termining extremal graphs for a given graph invariant in a set of graphs with prescribed properties. In the context of the theory of graph spectra, the invari- ant in question is a fixed eigenvalue of a matrix associated with a graph or a spectral invariant based on a number of graph eigenvalues (like the graph en- ergy). Extremal graphs for a given spectral invariant in various sets of graphs are widely considered. The terminology and notation are mainly taken from [98, 102], and they can also be found in similar literature. However, since there is some overlap in the wider notation used, we have made some small adjustments for this book only. The author is grateful to Dragoš Cvetković and Vladimir Nikiforov, who
- 18. Preface xi read themanuscript and gave valuable suggestions. In addition, these col- leagues – together with Kinkar Chandra Das, Martin Hasler, and Slobodan K. Simić – gave permission to use some of their proofs with no significant change. Finally, Sarah Lewis helped with correcting language and technical errors, which is much appreciated.
- 20. 1 Introduction In order tomake the reading of this book easier, in Section 1.1 we give a survey of the main graph-theoretic terminology and notation. Section 1.2 deals with matrix theory and graph spectra. In Section 1.3 we emphasize some more specific results of the theory of graph spectra that will frequently be used. Once we have fixed the notation and given all the necessary results, in Section 1.4 we say more about the applications of the theory of graph spectra and give some details related to the content of the book. 1.1 Graph-theoretic notions Let G be a finite undirected graph without loops or multiple edges on n vertices labelled 1,2,...,n. We denote the set of vertices of G by V (or V(G)). We say that two vertices i and j are adjacent (or neighbours) if they are joined by an edge and we write i ∼ j. We denote the set of edges of G by E (or E(G)), where an edge ij belongs to E if and only if i ∼ j. In this case we say that the edge ij is incident with vertices i and j. A graph consisting of a single vertex is called the trivial graph. Two edges are said to be adjacent if they are incident with a common vertex. Non-adjacent edges are said to be mutually independent. The number of vertices n and edges m in a graph are called the order and size, respectively. Two graphs G and H are said to be isomorphic if there is a bijection between V(G) and V(H) which preserves the adjacency of their vertices. The fact that G and H are isomorphic we denote by G ∼ = H, but we also use the simple notation G = H. A graph is asymmetric if the only permutation of its vertices which preserves their adjacency is the identity mapping. We say that G is the unique graph satisfying given properties if and only if any other graph with the same properties is isomorphic to G. 1
- 21. 2 Introduction A graphH obtained from a given graph G by deleting some vertices (to- gether with their edges incident) is called an induced subgraph of G. In this case, we also say that H is induced in G, and that G is a supergraph of H. We say that a graph G is H-free if it does not contain H as an induced subgraph. A subgraph of G is any graph H satisfying V(H) ⊆ V(G) and E(H) ⊆ E(G). If V(H) = V(G), H is called a spanning subgraph of G. If U ⊂ V(G), then we write G[U] to denote the induced subgraph of G with vertex set U and two vertices being adjacent if and only if they are adjacent in G. Similarly, an induced subgraph of G obtained by deleting a set of vertices V ⊆ V(G) is denoted by G−V (rather than G[V(G)V ]). If V consists of a single vertex v, we simply write G−v (instead of G− {v}). Similarly, G−E and G − e designate the deletion of a subset of edges E and a single edge e, respectively. By G+e we denote a graph obtained from G by inserting a single edge. If V1 and V2 are disjoint subsets of V(G), then m(V1) and m(V1,V2) stand for the number of edges in G[V1] and the number of edges with one end in V1 and the other in V2, respectively. The degree du of a vertex u (in a graph G) is the number of edges incident with it. In particular, the minimal and the maximal vertex degrees are denoted by δ and Δ, respectively. We say that a graph G is regular of degree r (or rG) if all its vertices have degree r. If so, then we usually say that G is r-regular. The complete graph on n vertices, Kn, is a graph whose every pair of vertices is joined by an edge. A regular graph of degree 3 is called a cubic graph. The unique (2n−2)-regular graph on 2n (n ≥ 1) vertices is called a cocktail party, and is denoted by CP(n). Obviously, it is an (n−1)-regular graph. A bidegreed graph has exactly two distinct vertex degrees. The edge degree of an edge uv is defined as du + dv − 2 (i.e., it is the number of edges that have a common vertex with uv). The set of neighbours (or the open neighbourhood) of a vertex u is denoted by N(u). The closed neighbourhood of u is denoted by N[u] (= {u} ∪ N(u)). The average degree of vertices in N(u) is denoted by mu, and it is also called the average 2-degree of u. A graph is said to be properly coloured if each vertex is coloured so that adjacent vertices have different colours. G is k-colourable if it can be properly coloured by k colours. The chromatic number χ is k if G is k-colourable and not (k − 1)-colourable. G is called bipartite if its chromatic number is 1 or 2. The vertex set of a bipartite graph can be partitioned into two parts (or colour classes) X and Y such that every edge of G joins a vertex in X with a vertex in Y. A graph is called complete bipartite if every vertex in one part is adjacent to every vertex in the other part. If |X| = n1 and |Y| = n2, the complete bipartite
- 22. 1.1 Graph-theoretic notions3 graph is denoted by Kn1,n2 . In particular, if n1 = 1, it is called a star. More generally, a k-partite graph is a graph whose set of vertices is discomposed into k disjoint sets such that no two vertices within the same set are adjacent. If there are n1,n2,...,nk vertices in the k sets, and if each two vertices which belong to different sets are adjacent, the graph is called complete k-partite (or simply complete multipartite) and denoted by Kn1,n2,...,nk . A graph is called semiregular bipartite if it is bipartite and the vertices be- longing to the same part have equal degree. If the corresponding vertex degrees are, say, r and s, the graph is referred to as (r,s)-semiregular bipartite. A vertex of degree 1 (in a graph G) is called an endvertex or pendant vertex. The edge incident with such a vertex is a pendant edge. A k-walk (or simply walk) in a graph G is an alternative sequence of vertices and edges v1,e1,v2,e2,...,ek−1,vk such that each edge ei is incident with vi and vi+1 (1 ≤ i ≤ k − 1). The walk is closed if v1 coincides with vk. The number of k-walks is denoted by wk. Similarly, the number of k-walks starting with u (resp. starting with u and ending with v) is denoted by wk(u) (resp. wk(u,v)). If all vertices of a walk are distinct, it is called a path. A graph which is itself a path on n vertices is denoted by Pn. An endvertex of Pn is often called an end of Pn. By joining the ends of Pn by an edge we get a cycle Cn. In particular, C3 is called a triangle and C4 is called a quadrangle. The number of triangles in a graph G is denoted by t(G). The length of a path Pn or a cycle Cn is equal to the number of edges contained in it. A graph is Hamiltonian if it contains a spanning subgraph which is a cycle, while any such cycle is referred to as a Hamiltonian cycle. We say that a graph G is connected if every two distinct vertices are the ends of at least one path in G. Otherwise, G is disconnected and its maximal connected induced subgraphs are called the components of G. A component consisting of a single vertex is called an isolated vertex (or trivial component) of G. A graph is totally disconnected if it consists entirely of isolated vertices. If G has exactly one non-trivial component, this component is called the dom- inant component. The distance d(u,v) between the vertices u and v is the length of the shortest path between u and v, and the girth gr(G) is the length of the shortest cycle induced in G. The diameter D of a graph G is the longest distance between two vertices of G. A shortest path between any pair of vertices u and v such that d(u,v) = D is called a diametral path. A connected graph G whose number of edges m equals n−1 is called a tree. Furthermore, if m = n−1+k (k ≥ 1), G is said to be k-cyclic. For k = 1, the corresponding graph is called unicyclic; for k = 2, it is called
- 23. 4 Introduction bicyclic. Clearly,any unicyclic graph contains a unique cycle as an induced subgraph. If this cycle has odd length then the graph is said to be odd unicyclic. Any complete induced subgraph of a graph G is called a clique. The clique number ω is the number of vertices in the largest clique of G. Similarly, any totally disconnected induced subgraph is called a co-clique. The vertices of a co-clique make an independent set of vertices of G, and the number of vertices in the largest independent set is called the independence number, denoted by α. A matching in G is a set of edges without common vertices. A matching is perfect if each vertex of G is incident with an edge from the matching. The matching number μ is the maximal size of a matching in G. A vertex (resp. edge) cover of a graph G is a set of vertices (resp. edges) such that each edge (resp. vertex) of G is incident with at least one vertex (resp. edge) of the set. The vertex (resp. edge) cover number of G, denoted by β (resp. β), is the minimum of the cardinalities of all vertex (resp. edge) covers. A dominating set for a graph G is a subset D of V(G) such that every vertex not in D is adjacent to at least one vertex in D. The domination number ϕ is the number of vertices in a smallest dominating set for G. A cut vertex (resp. cut edge) of a connected graph is any vertex (resp. edge) whose removal yields a disconnected graph. The vertex (resp. edge) connec- tivity, denoted by cv (resp. ce), of a connected graph is the minimal number of vertices (resp. edges) whose removal gives a disconnected graph. A rooted graph is a graph in which one vertex has been distinguished as the root. A pendant vertex of a rooted tree is often called a terminal vertex. For two graphs G and H we define G ∪ H to be their disjoint union.1 In addition, we use kG to denote the disjoint union of k copies of G. The join G∇H is the graph obtained by joining every vertex of G with every vertex of H. In particular, K1∇G is called the cone over G. The complement of a graph G is a graph G with the same vertex set as G, in which any two distinct vertices are adjacent if and only if they are non-adjacent in G. 1 With no confusion, we use the same symbol to denote the union of two sets. In addition, will stand for the union of disjoint sets.
- 24. 1.1 Graph-theoretic notions5 1.1.1 Some graphs The vertex with maximal degree in the star K1,n is called the centre of the star. The double star DS(n1,n2) is a graph obtained from the stars K1,n1−1 and K1,n2−1 by inserting an edge between their centres. A starlike tree Si1,i2,...,ik is a tree with exactly one vertex of degree greater than two such that the removal of this vertex gives rise to paths Pi1, Pi2 ,...,Pik . For k = 3, the corresponding starlike tree is often called a T-shape tree. A caterpillar is a tree in which the removal of all pendant vertices gives a path. Let the vertices of a path Pk (k ≥ 3) be labelled 1,2,...,k (in natural order), then T(m2,m3,...,mk−1) denotes the caterpillar obtained by attaching mi pendant vertices at the ith vertex of Pk (2 ≤ i ≤ k − 1). If a caterpillar is obtained by attaching just a few pendant vertices at the same path, we use the shorter notation T i1,i2,...,il n , where n denotes the order and ij (1 ≤ j ≤ l) indicates attaching a pendant vertex at the vertex labelled ij (2 ≤ ij ≤ k − 1). A closed caterpillar is a unicyclic graph in which the removal of all pendant vertices gives a cycle. An open quipu is a tree with maximal vertex degree 3 such that all vertices of degree 3 lie on a path. A closed quipu is a unicyclic graph with maximal vertex degree 3 such that all vertices of degree 3 lie on a cycle. A cactus is a connected graph G such that any two cycles induced in G have at most one common vertex. The comet C(k,l) is a tree obtained by attaching k pendant vertices at one end of the path Pl. The double comet DC(k,l) is a tree obtained by attaching k pendant vertices at one end of the path Pl and another k pendant vertices at the other end of the same path. The kite K(k,l) is a graph obtained by identifying one end of the path Pl+1 with a vertex of the complete graph Kk. The double kite DK(k,l) is a graph obtained by identifying one end of the path Pl+2 with a vertex of the complete graph Kk and the other end of the same path with a vertex of another complete graph Kk. The pineapple P(k,l) is a graph obtained by attaching l pendant vertices at a vertex of Kk. C(k,l), K(k,l), and P(k,l) have k + l vertices, while DC(k,l) and DK(k,l) have 2k + l vertices. Let GD = GD(n1,n2,...,nD+1) denote the graph defined as follows:V(GD ) = D+1 i=1 Vi, where GD[Vi] ∼ = Kni (1 ≤ i ≤ D+1) and
- 25. 6 Introduction GD [Vi ∪Vj]∼ = Kni+nj , if |i− j| = 1, Kni ∪Knj , otherwise. The graph GD consists of a chain of D + 1 cliques Kn1 ,Kn2 ,...,KnD+1 , where neighbouring cliques are fully interconnected (i.e., each vertex in one is ad- jacent to all vertices in the other). According to this, we name this graph the clique chain graph. Its order is n = ∑D+1 i=1 ni. Similarly, let GD ∗ = GD ∗ (n1,n2,...,nD+1) denote the graph of the same order defined as follows: V(GD ∗ ) = D+1 i=1 Vi, where GD ∗ [Vi] ∼ = niK1 (1 ≤ i ≤ D+1) and GD ∗ [Vi ∪Vj] ∼ = Kni,nj , if |i− j| = 1, (ni +nj)K1, otherwise. This graph consists of a chain of D + 1 co-cliques n1K1,n2K1,...,nD+1K1, where neighbouring co-cliques are fully interconnected. We name it the co- clique chain graph. Observe that GD ∗ is bipartite. Recall that a multigraph includes the possible existence of multiple edges between any two vertices or loops (i.e. edges with both endvertices identical). We say that a petal is added to a graph when we add a pendant vertex and then duplicate the edge incident with it. The line graph2 Line(G) of a multigraph G is the graph whose vertices are the edges of G, with two vertices adjacent whenever the corresponding edges have exactly one common vertex. Let G be a graph with vertex set V = {v1,v2,...,vn}, and let a1,a2,...,an be non-negative integers. The generalized line graph Line(G;a1,a2,...,an) is the graph Line( G), where G is the multigraph G(a1,a2,...,an) obtained from G by adding ai petals at vertex vi (1 ≤ i ≤ n). We introduce two classes of graphs called nested split graphs and double nested graphs. For these two classes of graphs we use the common name nested graphs. A nested split graph (NSG for short) is a graph which does not contain any of the graphs P4,C4 or 2K2 as an induced subgraph. This name is derived from its structure; it is also called a threshold graph (for more details, see [409]). We describe the structure of connected NSGs. The vertex set of any such graph consists of a co-clique and a clique, where both the co-clique and the clique are partitioned into h cells U1,U2,...,Uh and V1,V2,...,Vh, respectively. Then 2 The line graph is often denoted by L(G), but in this book L(G) is reserved for the Laplacian matrix (see Section 1.2).
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