Vertex-connectivity and eigenvalues of graphs

Hong, Zhen-Mu; Xia, Zheng-Jiang; Lai, Hong‐Jian

doi:10.1016/j.laa.2019.05.030

Cited by 10 publications

(6 citation statements)

References 15 publications

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“…Proposition 1.3. [25,Liu] Let k be an integer and G be a d-regular graph of order n with [20] found their bound for λ 2 (G) improving the previous two ones.…”

Section: Introductionmentioning

confidence: 88%

A best bound for $λ_2(G)$ to guarantee $κ(G) \geq 2$

Zhang¹,

Wang²

2021

Preprint

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Let G be a connected d-regular graph with a given order and the second largest eigenvalue λ 2 (G). Mohar and O (private communication) asked a challenging problem: what is the best upper bound for λ 2 (G) which guarantees that κ(G) ≥ t + 1, where 1 ≤ t ≤ d − 1 and κ(G) is the vertex-connectivity of G, which was also mentioned by Cioabȃ. As a starting point, we solve this problem in the case t = 1, and characterize all families of extremal graphs.

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“…Proposition 1.3. [25,Liu] Let k be an integer and G be a d-regular graph of order n with [20] found their bound for λ 2 (G) improving the previous two ones.…”

Section: Introductionmentioning

confidence: 88%

A best bound for $λ_2(G)$ to guarantee $κ(G) \geq 2$

Zhang¹,

Wang²

2021

Preprint

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“…From then on, the edge-connectivity problem has been intensively studied by many researchers, as found in [1,6,7,11,12,16,17,18,19,21,22], among others. For the vertex-connectivity of graphs, one can refer to [1,14,20,23]. In [1] (i…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.7 (Hong et al [14]) Let k be an integer and G be a simple graph of order n with maximum degree ∆ and minimum degree…”

Section: Introductionmentioning

confidence: 99%

Connectivity and eigenvalues of graphs with given girth or clique number

Hong

Lai

Xia

2020

Preprint

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Let κ ′ (G), κ(G), µ n−1 (G) and µ 1 (G) denote the edge-connectivity, vertex-connectivity, the algebraic connectivity and the Laplacian spectral radius of G, respectively. In this paper, we prove that for integers k ≥ 2 and r ≥ 2, and any simple graph G of order n with minimum degree δ ≥ k, girth g ≥ 3 and clique number ω(G) ≤ r, the edge-connectivity, where N (δ, g) is the Moore bound on the smallest possible number of vertices such that there exists a δ-regular simple graph with girth g, and ϕ(δ, r) = max{δ + 1, ⌊ rδ r−1 ⌋}. Analogue results involving µ n−1 (G) and µ 1 (G) µ n−1 (G) to characterize vertex-connectivity of graphs with fixed girth and clique number are also presented. Former results in [

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“…Motivated by the results of Liu et al [20]. Hong et al [13] found better bounds on λ 2 (G), µ 2 (G), or q 2 (G) which guarantee that κ(G) ≥ k.…”

mentioning

confidence: 99%

“…Theorem 1.9. ( [13]) Let G be a simple graph of order n with maximum degree ∆ and minimum degree δ ≥ k for any integer k > 0. Let…”

mentioning

confidence: 99%

The vertex connectivity and the third largest eigenvalue in regular (multi-)graphs

Ma,

Wang,

2024

ELA

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Let $G$ be a simple graph or a multigraph. The vertex connectivity $\kappa(G)$ of $G$ is the minimum size of a vertex set $S$ such that $G-S$ is disconnected or has only one vertex. We denote by $\lambda_{3}(G)$ the third largest eigenvalue of the adjacency matrix of $G$. In this paper, we present an upper bound for $\lambda_{3}(G)$ in a $d$-regular (multi-)graph $G$ which guarantees that $\kappa(G)\geq t+1$, which is based on the result of Abiad et al. [Spectral bounds for the connectivity of regular graphs with given order. Electron. J. Linear Algebra 34:428-443, 2018]. Furthermore, we improve the upper bound for $\lambda_{3}(G)$ in a $d$-regular multigraph which assures that $\kappa(G)\geq 2$.

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Vertex-connectivity and eigenvalues of graphs

Cited by 10 publications

References 15 publications

A best bound for $λ_2(G)$ to guarantee $κ(G) \geq 2$

A best bound for $λ_2(G)$ to guarantee $κ(G) \geq 2$

Connectivity and eigenvalues of graphs with given girth or clique number

The vertex connectivity and the third largest eigenvalue in regular (multi-)graphs

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