Distance spectral radius for a graph to be k-critical with respect to [1,b]-odd factor

TL;DR

This paper establishes an upper bound on the distance spectral radius of a connected graph that ensures the graph is k-critical with respect to the existence of a specific type of odd factor, advancing understanding of spectral conditions for graph factors.

Contribution

It introduces a new spectral bound on the distance matrix that guarantees a graph's k-critical property related to [1,b]-odd factors, a novel criterion in spectral graph theory.

Findings

Derived an explicit upper bound for the distance spectral radius.

Proved the bound guarantees the k-critical property.

Enhanced understanding of spectral conditions for graph factors.

Abstract

Let $G$ be a connected graph, and let $b$ and $k$ be two positive integers with $b\equiv1$ (mod 2). A $[1,b]$-odd factor of $G$ is a spanning subgraph $F$ of $G$ with $d_F(v)\equiv1$ (mod 2) and $1\leq d_F(v)\leq b$ for every $v\in V(G)$. A graph $G$ is called $k$-critical with respect to $[1,b]$-odd factor if $G-X$ contains a $[1,b]$-odd factor for every $X\subseteq V(G)$ with $|X|=k$. Let $\mathcal{D}(G)$ denote the distance matrix of $G$. The largest eigenvalue of $\mathcal{D}(G)$, denoted by $\mu(G)$, is called the distance spectral radius of $G$. In this paper, we prove an upper bound for $\mu(G)$ in a connected graph $G$ which guarantees $G$ to be $k$-critical with respect to $[1,b]$-odd factor.