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On the multiplicity of Laplacian eigenvalues for unicyclic graphs

Fei WenQiongxiang Huang — 2022

Czechoslovak Mathematical Journal

Let G be a connected graph of order n and U a unicyclic graph with the same order. We firstly give a sharp bound for m G ( μ ) , the multiplicity of a Laplacian eigenvalue μ of G . As a straightforward result, m U ( 1 ) n - 2 . We then provide two graph operations (i.e., grafting and shifting) on graph G for which the value of m G ( 1 ) is nondecreasing. As applications, we get the distribution of m U ( 1 ) for unicyclic graphs on n vertices. Moreover, for the two largest possible values of m U ( 1 ) { n - 5 , n - 3 } , the corresponding graphs U are completely...

Spectral characterization of multicone graphs

Jianfeng WangHaixing ZhaoQiongxiang Huang — 2012

Czechoslovak Mathematical Journal

A multicone graph is defined to be the join of a clique and a regular graph. Based on Zhou and Cho's result [B. Zhou, H. H. Cho, Remarks on spectral radius and Laplacian eigenvalues of a graph, Czech. Math. J. 55 (130) (2005), 781–790], the spectral characterization of multicone graphs is investigated. Particularly, we determine a necessary and sufficient condition for two multicone graphs to be cospectral graphs and investigate the structures of graphs cospectral to a multicone graph. Additionally,...

The chromatic equivalence class of graph B n - 6 , 1 , 2 ¯

Jianfeng WangQiongxiang HuangChengfu YeRuying Liu — 2008

Discussiones Mathematicae Graph Theory

By h(G,x) and P(G,λ) we denote the adjoint polynomial and the chromatic polynomial of graph G, respectively. A new invariant of graph G, which is the fourth character R₄(G), is given in this paper. Using the properties of the adjoint polynomials, the adjoint equivalence class of graph B n - 6 , 1 , 2 is determined, which can be regarded as the continuance of the paper written by Wang et al. [J. Wang, R. Liu, C. Ye and Q. Huang, A complete solution to the chromatic equivalence class of graph B n - 7 , 1 , 3 ¯ , Discrete Math....

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