A unified combinatorial view beyond some spectral properties

TL;DR

This paper introduces a new graph class called weakly $(n,eta)$-graphs and $(n,eta)$-graphs, providing bounds on their matching numbers, existence of perfect matchings, and toughness, extending spectral property analyses.

Contribution

It defines new graph classes inspired by spectral properties and jumbled graphs, and establishes bounds on matchings and toughness for these classes, broadening spectral graph theory insights.

Findings

Matching number bounds for weakly and $(n,eta)$-graphs.

Existence of perfect and fractional perfect matchings.

Lower bounds on graph toughness for these classes.

Abstract

Let $\beta>0$. Motivated by jumbled graphs defined by Thomason, the celebrated expander mixing lemma and Haemers's vertex separation inequality, we define that a graph $G$ with $n$ vertices is a weakly $(n,\beta)$-graph if $\frac{|X| |Y|}{(n-|X|)(n-|Y|)} \le \beta^2$ holds for every pair of disjoint proper subsets $X, Y$ of $V(G)$ with no edge between $X$ and $Y$, and it is an $(n,\beta)$-graph if in addition $X$ and $Y$ are not necessarily disjoint. Our main results include the following. (i) For any weakly $(n,\beta)$-graph $G$, the matching number $\alpha'(G)\ge \min\left\{\frac{1-\beta}{1+\beta},\, \frac{1}{2}\right\}\cdot (n-1).$ If in addition $G$ is a $(U, W)$-bipartite graph with $|W|\ge t|U|$ where $t\ge 1$, then $\alpha'(G)\ge \min\{t(1-2\beta^2),1\}\cdot |U|$. (ii) For any $(n,\beta)$-graph $G$, $\alpha'(G)\ge \min\left\{\frac{2-\beta}{2(1+\beta)},\, \frac{1}{2}\right\}\cdot (n-1).$ If in addition $G$ is a $(U, W)$-bipartite graph with $|W|\ge |U|$ and no isolated vertices, then $\alpha'(G)\ge \min\{1/\beta^{2},1\}\cdot |U|$. (iii) If $G$ is a weakly $(n,\beta)$-graph for $0<\beta\le 1/3$ or an $(n,\beta)$-graph for $0<\beta\le 1/2$, then $G$ has a fractional perfect matching. In addition, $G$ has a perfect matching when $n$ is even and $G$ is factor-critical when $n$ is odd. (iv) For any connected $(n,\beta)$-graph $G$, the toughness $t(G)\ge \frac{1-\beta}{\beta}$. For any connected weakly $(n,\beta)$-graph $G$, $t(G)> \frac{5(1-\beta)}{11\beta}$ and if $n$ is large enough, then $t(G) >\left(\frac{1}{2}-\varepsilon\right)\frac{1-\beta}{\beta}$ for any $\varepsilon >0$.