On the largest eigenvalue of non-regular graphs

Abstract

We study the spectral radius of connected non-regular graphs. Let λ 1 (n, Δ) be the maximum spectral radius among all connected non-regular graphs with n vertices and maximum degree Δ. We prove that Δ − λ 1 (n, Δ) = Θ(Δ/n 2 ). This improves two recent results by Stevanović and Zhang, respectively.

The Laplacian spectral radius of graphs

“…, 1/ √ n) as its Perron eigenvector. Recently, Stevanović [10], Zhang [12], Liu et al [5], and Cioabȃ [2] studied the spectral radius of irregular graphs. The current best result is due to Cioabȃ as follows: For an irregular graph G of order n with maximum degree ∆ and diameter D, then λ(G) < ∆ − 1/nD.…”

Eigenvalues of subgraphs of the cube

“…In the last few decades much research has been done on spectra of graphs, i.e. the eigenvalues of the adjacency matrices of graphs; see Finck and Grohmann [10], Hoffman [16,17], Nosal [25], Cvetković, Doob and Sachs [7], Neumaier [20], Brigham and Dutton [3,4], Brualdi and Hoffman [5], Stanley [30], Shearer [29], Powers [26], Favaron, Mahéo and Saclé [8,9], Hong [18], Liu, Shen and Wang [19], Nikiforov [22,23,24,21], and Cvetković, Rowlinson and Simić [6] for a small selection of relevant publications. Perhaps the most basic property of the spectrum of a graph is its radius, i.e.…”

Maximum spectral radius of graphs with given connectivity, minimum degree and independence number

“…Feng, Yu and Zhang [6] studied this problem for C to be the set of graphs on n vertices with matching number β. The relation of spectral radius and maximum degree of graphs are characterized in [4,5] and by Liu, Shen and Wang in [11]. Also Bhattacharya, Friedland and Peled [2] studied the maximum value of the largest eigenvalue for simple bipartite graphs in term of the number of edges and the number of vertices on each side of the bipartition.…”