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Extremal distance spectra of graphs and essential connectivity

Daoxia Zhang, Dan Li, Wenxiu Ding

TL;DR

This work studies how essential connectivity $κ'(G)$ influences the distance spectral radius of graphs and digraphs by identifying extremal structures under fixed order, and, when applicable, fixed minimum degree. It determines that, for graphs of order $n$ with a given $κ'$, the minimum $λ_{1}(D(G))$ is achieved by the join $K_{κ'} abla (K_2 6cup K_{n-κ'-2})$, and extends this to the case of fixed $κ'$ and $δ$ with an explicit construction $G_{n}^{κ',δ}$ for which equality holds. The analogous results for strongly connected digraphs yield extremal forms in the family $ ightarrow{G}_{n}^{k,n_{1}}$, with the minimum occurring at $n_{1}=2$ or $n_{1}=n-k-2$, giving $ ightarrow{G}_{n}^{k,2}$ or $ ightarrow{G}_{n}^{k,n-k-2}$. The methods combine structural analyses around essential vertex-cuts, joins of complete graphs with unions of cliques, and Perron-vector techniques to compare distances and derive exact extremal graphs and their distance spectral radii. These findings advance our understanding of distance-spectral extremal problems under connectivity constraints and have implications for network robustness assessment.

Abstract

A graph is non-trivial if it contains at least one nonloop edge. The essential connectivity of $G$, denoted by $κ'(G)$, is the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two components are non-trivial. In this paper, we determine the $n$-vertex graph of given essential connectivity with minimum distance spectral radius. We also characterize the extremal graphs attaining the minimum distance spectral radius among all connected graphs with fixed essential connectivity and minimum degree. Furthermore, we characterize the extremal digraphs with minimum distance spectral radius among the strongly connected digraphs with given essential connectivity.

Extremal distance spectra of graphs and essential connectivity

TL;DR

This work studies how essential connectivity influences the distance spectral radius of graphs and digraphs by identifying extremal structures under fixed order, and, when applicable, fixed minimum degree. It determines that, for graphs of order with a given , the minimum is achieved by the join , and extends this to the case of fixed and with an explicit construction for which equality holds. The analogous results for strongly connected digraphs yield extremal forms in the family , with the minimum occurring at or , giving or . The methods combine structural analyses around essential vertex-cuts, joins of complete graphs with unions of cliques, and Perron-vector techniques to compare distances and derive exact extremal graphs and their distance spectral radii. These findings advance our understanding of distance-spectral extremal problems under connectivity constraints and have implications for network robustness assessment.

Abstract

A graph is non-trivial if it contains at least one nonloop edge. The essential connectivity of , denoted by , is the minimum number of vertices of whose removal produces a disconnected graph with at least two components are non-trivial. In this paper, we determine the -vertex graph of given essential connectivity with minimum distance spectral radius. We also characterize the extremal graphs attaining the minimum distance spectral radius among all connected graphs with fixed essential connectivity and minimum degree. Furthermore, we characterize the extremal digraphs with minimum distance spectral radius among the strongly connected digraphs with given essential connectivity.
Paper Structure (3 sections, 8 theorems, 60 equations, 5 figures)

This paper contains 3 sections, 8 theorems, 60 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1.1']} and \ref{['thm1.2']}
  3. Proof of Theorem \ref{['thm1.3']}

Key Result

Theorem 1

Let $G$ be a connected graph of order $n\geq\kappa'+4$ with essential connectivity $\kappa'$. Then with equality if and only if $G\cong K_{\kappa'}\vee(K_{2}\cup K_{n-\kappa'-2})$.

Figures (5)

  • Figure 1: The digraph $\overrightarrow{G}_{n}^{k,n_{1}}$.
  • Figure 2: The graphs $G$ and $G_{1}$.
  • Figure 3: The graphs $G$ and $G_{2}$.
  • Figure 4: The graphs $G$ and $G_{3}$.
  • Figure 5: The graphs $G$ and $G_{4}$.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Claim 1
  • Claim 2
  • Claim 3
  • Lemma 3
  • Lemma 4
  • ...and 1 more