On the distance spectral radius, fractional matching and factors of graphs with given minimum degree
Zengzhao Xu, Weige Xi, Ligong Wang
TL;DR
The paper investigates how the distance spectral radius μ(G) constrains fractional matchings and the existence of specific graph factors in connected graphs with prescribed minimum degree δ. It derives sharp upper bounds on μ(G) that force the fractional matching number α_f(G) to exceed (n−k)/2 and extends these spectral bounds to guarantee the presence of {K2, {C_k}}-factors and {K_{1,1},...,K_{1,k}}-factors, identifying extremal configurations. The extremal graphs are explicit joins of complete graphs with independent sets, notably the structure $K_δ \\\vee (K_{n-2δ-k}+(δ+k)K_1)$. The results employ equitable quotient matrices for the distance matrix and spanning-subgraph comparisons to establish tight, geometry-driven thresholds with clear necessary and sufficient conditions.
Abstract
A fractional matching of $G$ is a function $f: E(G)\to [0,1]$ such that $\sum_{e\in E_G(v_i)}f(e)\le 1$ for any $v_i\in V(G)$, where $E_G(v_i)=\{e: e\in E(G) \ \textrm{and}\ e \ \textrm{is incident with} \ v_i\}$. Let $α_f(G)$ denote the fractional matching number of $G$, which is defined as $α_f(G)=\max\{\sum_{e\in E(G)}f(e): f\ \textrm{is a fractional matching of} \ G\}$. Let $\{G_1,G_2,G_3,\dots\}$ be a set of graphs, a $\{G_1,G_2,G_3,\dots\}$-factor of a graph $G$ is a spanning subgraph of $G$ such that each component of which is isomorphic to one of $\{G_1,G_2,G_3,\dots\}$. In this paper, we first establish a sharp upper bound for the distance spectral radius to guarantee that $α_f(G)>\frac{n-k}{2}$ in a graph $G$ of order $n$ with given minimum degree, where $0<k<n$ is an integer. Then we give a sharp upper bound on the distance spectral radius of a graph $G$ with given minimum degree $δ$ to ensure that $G$ has a $\{K_2, \{C_k\}\}$-factor, where $3\le k<+\infty$ is an integer. Moreover, we obtain a sharp upper bound on the distance spectral radius for the existence of a $\{K_{1,1},K_{1,2},\dots,K_{1,k}\}$-factor with $2\le k<+\infty$ in a graph $G$ with given minimum degree.