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1. *Corresponding Author: Rao Li, Email: raol@usca.edu
REVIEW ARTICLE
Available Online at www.ajms.in
Asian Journal of Mathematical Sciences 2017; 1(2):65-66
Signless Laplacian Spread and Hamiltonicity of Graphs
1
Rao Li*
1
*Dept. of mathematical sciences University of South Carolina Aiken, Aiken, SC 29801
Received on: 18/02/2017, Revised on: 06/03/2017, Accepted on: 14/03/2017
ABSTRACT
The signless Laplacian spread of a graph G is the difference between the largest and smallest signless
Lapacian eigenvalues of G. Using a result on the signless Laplacian spread obtained by Liu and Liu in
[3], we in this note present a sufficient condition based on the signless Laplacian spread for the
Hamiltonicity of graphs. 2010 Mathematics Subject Classification: 05C50, 05C45.
Keywords : signless Laplacian spread, Hamiltonian properties
INTRODUCTION
We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not
defined here follow those in [1]. For a graph G = (V (G), E(G)), we use n and e to denote its order |V (G)|
and size |E(G)|, respectively. For a subset V1 of V , its average degree is defined as P
v∈V1 d(v)/|V1|. Let
D(G) be a diagonal matrix such that its diagonal entries are the degrees of vertices in a graph G. The
signless Laplacian matrix of a graph G, denoted Q(G), is defined as D(G)+A(G), where A(G) is the
adjacency matrix of G. The eigenvalues of Q(G) are called the signless Laplacian eigenvalues of G. The
signless Laplacian spread of a graph G, denoted by SQ(G), was introduced by Liu and Liu in [3]. It is
defined as the difference between the largest and smallest signless Laplacian eigenvalues of G. A cycle C
in a graph G is called a Hamiltonian cycle of G if C contains all the vertices of G. A graph G is called
Hamiltonian if G has a Hamiltonian cycle. The purpose of this note is to present a sufficient condition
based on the signless Laplacian spread for the Hamiltonicity of graphs. The main result is as follows.
Theorem 1. Let G be a graph of order n, size e, and connectivity κ(κ ≥ 2).
If
,
then G is Hamiltonian.
In order to prove Theorems 1, we need the following result as our lemma. Lemma 1 below is Theorem
2.3 on Page 509 in [3].
Lemma 1. Let G be a connected graph on n ≥ 2 vertices with e edges. Suppose G contains a nonempty
set of t independent vertices, the average degree of which is d0. Then
.
Proof of Theorem 1.
Let G be a graph satisfying the conditions in Theorem 1. Suppose that G is not Hamiltonian. Since κ ≥ 2,
G contains a cycle. Choose a longest cycle C in G and give an orientation on C. Since G is not
Hamiltonian, there exists a vertex x0 ∈ V (G)V (C). By Menger’s theorem, we can find s (s ≥ κ) pairwise

2. Li Rao et al. Signless Laplacian Spread and Hamiltonicity of Graphs
66
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disjoint (except for x0) paths P1, P2, ..., Ps between x0 and V (C). Let di be the end vertex of Pi on C, where
1 ≤ i ≤ s. We use xi to denote the successor of di along the orientation of C, where 1≤ i ≤ s. Then a
standard proof in Hamiltonian graph theory yields that S := {x0,x1,x2,...,xs} is independent (otherwise G
would have cycles which are longer than C). Set X := {x0,x1,x2,...,xκ}. Obviously, X is independent.
Following the proofs of the main theorem in [2], we define
A = V (G) −N(X), AC = A ∩ C, AR = A −C.
B = {v ∈ V (G) : there is a unique i such that vxi ∈ E(G)}.
D = V (G) −(A ∪ B).
From the proofs of the main theorem in [2], we have the following true propositions.
Proposition 1. X ⊂A. Therefore, x0 ∈ AR.
Proposition 2. Two vertices of X have no common neighbor in V (G) −V (C), that is, (V (G) −V (C) ∩ D
= ∅
.
Proposition 3. There are at most κ elements of D between two consecutive vertices of A on C.
From the Propositions 1 - 3, we have, for each i with 0 ≤ i ≤ κ, that
|N(X − {xi})| ≤ n − |A| − dB(xi).
Thus
Therefore
.
Hence the average degree davg of X is less than or equal to .
Applying Lemma 1 to the independent set X of κ + 1 vertices, we have that
,
a contradiction. This completes the proof of Theorem 1.
REFERENCES
1. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London and
Elsevier, New York (1976).
2. P. Fraisse, A new sufficient condition for Hamiltonian graphs, Journal of Graph Theory 10 (1986)
405 – 409.
3. M. Liu and B. Liu, The signless Laplacian spread, Linear Algebra and its Applications 432 (2010)
505 – 514.
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