Matching, odd $[1,b]$-factor and distance spectral radius of graphs with given some parameters

TL;DR

This paper investigates how the distance spectral radius $\mu(G)$ governs two classic graph properties: large matchings and odd $[1,b]$-factors. It derives sharp $\mu$-bounds for $n$-vertex $t$-connected graphs that force the matching number to exceed $(n-k)/2$, with a precisely characterized extremal graph $K_t \vee (K_{n+1-2t-k}+(t+k-1)K_1)$. It also presents sharp $\mu$-bounds for the existence of odd $[1,b]$-factors in graphs with a given minimum degree $\delta$, identifying the extremal join $K_{\delta} \vee (K_{n-(b+1)\delta-1}+(b\delta+1)K_1)$ and including a corollary for perfect matchings when $b=1$. The results extend the spectral-graph theory toolkit by linking distance spectrum to fundamental decomposition and spanning-factor properties, with exact extremal graphs fully characterized.

Abstract

For a connected graph $G$, let $μ(G)$ denote the distance spectral radius of $G$. A matching in a graph $G$ is a set of disjoint edges of $G$. The maximum size of a matching in $G$ is called the matching number of $G$, denoted by $α(G)$. An odd $[1, b]$-factor of a graph $G$ is a spanning subgraph $G_0$ such that the degree $d_{G_0}(v)$ of $v$ in $G_0$ is odd and $1\le d_{G_0}(v)\le b$ for every vertex $v\in V (G)$. In this paper, we give a sharp upper bound in terms of the distance spectral radius to guarantee $α(G)>\frac{n-k}{2}$ in an $n$-vertex $t$-connected graph $G$, where $2\le k \le n-2$ is an integer. We also present a sharp upper bound in terms of distance spectral radius for the existence of an odd $[1,b]$-factor in a graph with given minimum degree $δ$.

Figures (2)

• Figure 1: The extremal graph of Theorem 3.2.
• Figure 2: The extremal graph of Theorem 4.1.

Theorems & Definitions (17)

• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 3.1
• Theorem 3.2
• Lemma 3.3