Abstract:
We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial.

Abstract:
Two nonisomorphic graphs G and H are said to be matching equivalent if and only if G and H have the same matching polynomials. In this paper, some families matching equivalent graphs are constructed. In particular, a new method to construct cospectral forests is given.

Abstract:
In an earlier paper, the present authors (2013) introduced the altermatic number of graphs and used Tucker's Lemma, an equivalent combinatorial version of the Borsuk-Ulam Theorem, to show that the altermatic number is a lower bound for the chromatic number. A matching graph has the set of all matchings of a specified size of a graph as vertex set and two vertices are adjacent if the corresponding matchings are edge-disjoint. It is known that the Kneser graphs, the Schrijver graphs, and the permutation graphs can be represented by matching graphs. In this paper, as a generalization of the well-known result of Schrijver about the chromatic number of Schrijver graphs, we determine the chromatic number of a large family of matching graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching graphs in terms of the generalized Turan number of matchings.

Abstract:
Counter to the general notion that the regular tree is the worst case for decay of correlation between sets and nodes, we produce an example of a multi-spin interacting system which has uniqueness on the $d$-regular tree but does not have uniqueness on some infinite $d$-regular graphs.

Abstract:
The maximum matching problem on random graphs is studied analytically by the cavity method of statistical physics. When the average vertex degree \mth{c} is larger than \mth{2.7183}, groups of max-matching patterns which differ greatly from each other {\em gradually} emerge. An analytical expression for the max-matching size is also obtained, which agrees well with computer simulations. Discussion is made on this {\em continuous} glassy phase transition and the absence of such a glassy phase in the related minimum vertex covering problem.

Abstract:
Given a set $P$ of $n$ points in the plane, the order-$k$ Gabriel graph on $P$, denoted by $k$-$GG$, has an edge between two points $p$ and $q$ if and only if the closed disk with diameter $pq$ contains at most $k$ points of $P$, excluding $p$ and $q$. We study matching problems in $k$-$GG$ graphs. We show that a Euclidean bottleneck perfect matching of $P$ is contained in $10$-$GG$, but $8$-$GG$ may not have any Euclidean bottleneck perfect matching. In addition we show that $0$-$GG$ has a matching of size at least $\frac{n-1}{4}$ and this bound is tight. We also prove that $1$-$GG$ has a matching of size at least $\frac{2(n-1)}{5}$ and $2$-$GG$ has a perfect matching. Finally we consider the problem of blocking the edges of $k$-$GG$.

Abstract:
We study a competitive optimization version of $\alpha'(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be large; the other (Min) wants it to be small. The resulting sizes under optimal play when Max or Min starts are denoted $\Max(G)$ and $\Min(G)$, respectively. We show that always $|\Max(G)-\Min(G)|\le 1$. We obtain a sufficient condition for $\Max(G)=\alpha'(G)$ that is preserved under cartesian product. In general, $\Max(G)\ge \frac23\alpha'(G)$, with equality for many split graphs, while $\Max(G)\ge\frac34\alpha'(G)$ when $G$ is a forest. Whenever $G$ is a 3-regular $n$-vertex connected graph, $\Max(G) \ge n/3$, and there are such examples with $\Max(G)\le 7n/18$. For an $n$-vertex path or cycle, the answer is roughly $n/7$.

Abstract:
A graph is called matching covered if for its every edge there is a maximum matching containing it. It is shown that minimal matching covered graphs contain a perfect matching.

Abstract:
In the paper, we give a necessary and
sufficient condition of matching equivalence of two graphs with the maximum
matching root less than or equal to 2.

Abstract:
Gutman and Wagner proposed the concept of the matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let $G$ be a simple graph of order $n$ and $\mu_1,\mu_2,\ldots,\mu_n$ be the roots of its matching polynomial. The matching energy of $G$ is defined to be the sum of the absolute values of $\mu_{i}\ (i=1,2,\ldots,n)$. Gutman and Cvetkoi\'c determined the tricyclic graphs on $n$ vertices with maximal number of matchings by a computer search for small values of $n$ and by an induction argument for the rest. Based on this result, in this paper, we characterize the graphs with the maximal value of matching energy among all tricyclic graphs, and completely determine the tricyclic graphs with the maximal matching energy. We prove our result by using Coulson-type integral formula of matching energy, which is similar as the method to comparing the energies of two quasi-order incomparable graphs.