A unified combinatorial view beyond some spectral properties

Let \(\beta >0\). Motivated by the notion of jumbled graphs introduced by Thomason, the expander mixing lemma and Haemers’s vertex separation inequality, we say 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. It is an \((n,\beta )\)-graph if in addition X and Y are not necessarily disjoint. Using graph eigenvalues, we show that every graph can be an \((n,\beta )\)-graph and/or a weakly \((n,\beta )\)-graph for some specific value \(\beta \). For instances, the expander mixing lemma implies that a d-regular graph on n vertices with the second largest absolute eigenvalue at most \(\lambda \) is an \((n,\lambda /d)\)-graph, and Haemers’s vertex separation inequality implies that every graph is a weakly \((n,\beta )\)-graph with \(\beta \ge \frac{\mu _n - \mu _2}{\mu _n + \mu _2}\), where \(\mu _i\) denotes the i-th smallest Laplacian eigenvalue. This motivates us to study \((n,\beta )\)-graph and weakly \((n,\beta )\)-graph in general. 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 \(\varepsilon >0\). The results imply many old and new results in spectral graph theory, including several new lower bounds on matching number, fractional matching number and toughness from eigenvalues. In particular, we obtain a new lower bound on toughness via normalized Laplacian eigenvalues that extends a theorem originally conjectured by Brouwer from regular graphs to general graphs.