Elsevier

Discrete Mathematics

Volume 341, Issue 11, November 2018, Pages 2969-2976
Discrete Mathematics

Which cospectral graphs have same degree sequences

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Abstract

Two graphs are said to be L-cospectral (respectively, Q-cospectral) if they have the same (respectively, signless) Laplacian spectra, and a graph G is said to be LDS (respectively, QDS) if there does not exist other non-isomorphic graph H such that H and G are L-cospectral (respectively, Q-cospectral). Let d1(G)d2(G)dn(G) be the degree sequence of a graph G with n vertices. In this paper, we prove that except for two exceptions (respectively, the graphs with d1(G){4,5}), if H is L-cospectral (respectively, Q-cospectral) with a connected graph G and d2(G)=2, then H has the same degree sequence as G. A spider graph is a unicyclic graph obtained by attaching some paths to a common vertex of the cycle. As an application of our result, we show that every spider graph and its complement graph are both LDS, which extends the corresponding results of Haemers et al. (2008), Liu et al. (2011), Zhang et al. (2009) and Yu et al. (2014).

Keywords

(Signless) Laplacian spectrum
Determined by spectrum
Degree sequence
Q-cospectral
L-cospectral

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Supported by NSFC of China (No. 11571123), the Training Program for Outstanding Young Teachers in University of Guangdong Province (No. YQ2015027), the Guangdong Provincial Natural Science Foundation (No. 2015A030313377), Guangdong Engineering Research Center for Data Science and Guangdong Province Ordinary University Characteristic Innovation Project (No. 2017KTSCX020).