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Let ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n − 1 ≥ ρ n = 0 be the normalized Laplacian eigenvalues of a graph G with n vertices. Also, let χ and α be the chromatic number and the independence number of a graph G, respectively. In this paper, we discuss some properties of graphs with ρ 1 = χ / (χ − 1). In particular, we characterize all the graphs with ρ 1 = χ / (χ − 1) when the maximum degree is n−1. Moreover, we obtain an upper bound on the multiplicity of normalized Laplacian eigenvalues m G (ρ) in terms of n and α, and also characterize graphs for which the bound is attained. [ABSTRACT FROM AUTHOR]

Let , and be, respectively, the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix of graph G, where denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of and will be denoted by and , respectively. In this paper, we study the properties of the distance Laplacian eigenvalues and the distance signless Laplacian eigenvalues of graph G. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a connected graph with vertex set V(G) = {v1, v2,..., vn} and edge set E(G). Let D(G) be the distance matrix of G and λ1(D) ≥···≥ λn(D) be its distance spectrum. The distance between distance spectra of G and G' is defined by σ(G,G') = i=1∑n|λi(D(G)) - λi(D(G'))|. Define the cospectrality of G by cs(G) = min {i=1∑n|λi(D(G)) - λi(D(G'))|: G' not isomorphic to G}. Let csn = max{cs(G) : G a connected graph on n vertices}. In the paper, we obtain lower bounds on σ(G, Kn) and σ(G, Ka,b) for a+b = n. Furthermore, we give an upper bound on csn. [ABSTRACT FROM AUTHOR]

Letdenote the class ofc-cyclic graphs withnvertices, girthandpendant vertices. In this paper, we determine the unique extremal graph with largest signless Laplacian spectral radius and Laplacian spectral radius in the class of connectedc-cyclic graphs withvertices, girthgand at mostpendant vertices, respectively, and the unique extremal graph with largest signless Laplacian spectral radius ofwhenand, and we also identify the unique extremal graph with largest Laplacian spectral radius inin the caseand eitherandgis even orandgis odd. Our results extends the corresponding results of [Sci. Sin. Math. 2010;40:1017–1024, Electron. J. Combin. 2011; 18:p.183, Comput. Math. Appl. 2010;59:376–381, Electron. J. Linear Algebra. 2011;22:378–388 and J. Math. Res. Appl. 2014;34:379–391]. [ABSTRACT FROM AUTHOR]

In this article, we present lower bounds for the largest eigen value, the second largest eigenvalue and the sum of the two largest eigenvalues of the Laplacian matrix of a graph. [ABSTRACT FROM AUTHOR]