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

TL;DR

This work studies the distance spectral radius $μ(G)$ of a connected graph $G$ in relation to the existence of a $[1,b]$-odd factor after deleting any set of $k$ vertices. The authors prove Theorem 1.1, which gives an explicit upper bound on $μ(G)$—namely $μ(G) ≤ μ(K_{k+1} \vee (K_{n-k-b-2} \cup (b+1)K_1))$ under the order condition $n ≥ (b^{2}+2bk+5b+2k+4)/b$ and with $b$ odd—ensuring that $G$ is $k$-critical with respect to $[1,b]$-odd factor, except for two explicit graphs. The proof combines a detailed extremal-structure analysis, the equitable-partition/quotient-matrix framework, and a comparison against the extremal graph $G_* = K_{k+1} \vee (K_{n-k-b-2} \cup (b+1)K_1)$ to derive a contradiction in all non-exceptional cases. The result extends spectral-conditions-based criteria for the existence of graph factors to the distance spectrum, providing a practical tool for guaranteeing $[1,b]$-odd factors after vertex deletions via a computable bound on $μ(G)$.

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 $μ(G)$, is called the distance spectral radius of $G$. In this paper, we prove an upper bound for $μ(G)$ in a connected graph $G$ which guarantees $G$ to be $k$-critical with respect to $[1,b]$-odd factor.