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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 $m_G\left(\mu \right)$, the multiplicity of a Laplacian eigenvalue $\mu$ of $G$. As a straightforward result, $m_U\left(1\right)\le n-2$. We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_G\left(1\right)$ is nondecreasing. As applications, we get the distribution of $m_U\left(1\right)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_U\left(1\right)\in \{n-5,n-3\}$, the corresponding graphs $U$ are completely...

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,...

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 $\overline{B_{n-7,1,3}}$, Discrete Math....