Vertex-connectivity and eigenvalues of graphs with fixed girth

Ruifang Liu, Hong-Jian Lai, Yingzhi Tian and Yang Wu

Applied Mathematics and Computation, 2019, vol. 344-345, 141-149

Abstract: Let κ(G), g(G), δ(G) and Δ(G) denote the vertex-connectivity, the girth, the minimum degree and the maximum degree of a simple graph G, and let λi(G), μi(G) and qi(G) denote the ith largest adjacency eigenvalue, Lapalcian eigenvalue and signless Laplacian eigenvalue of G. We investigate functions f(δ, Δ, g, k) with Δ ≥ δ ≥ k ≥ 2 and g ≥ 3 such that any graph G satisfying λ2(G) < f(δ(G), Δ(G), g(G), k) has connectivity κ(G) ≥ k. Analogues results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues to describe connectivity of a graph are also presented. As corollaries, we show that for an integer k ≥ 2 and a simple graph G with n=|V(G)|, maximum degree Δ and minimum degree δ ≥ k, the connectivity κ(G) ≥ k if one of the following holds.(i)λ2(G)<δ−(k−1)Δn2(δ−k+2)(n−δ+k−2), or(ii)μn−1(G)>(k−1)Δn2(δ−k+2)(n−δ+k−2), or(iii)q2(G)<2δ−(k−1)Δn2(δ−k+2)(n−δ+k−2).

Keywords: Girth; Connectivity; Adjacency eigenvalue; Laplacian eigenvalue; Signless Laplacian eigenvalue; Quotient matrix (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:344-345:y:2019:i::p:141-149

DOI: 10.1016/j.amc.2018.09.053

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