In this paper, we obtain the upper and lower bounds on the eccen- tricity connectivity index of unicyclic graphs with perfect matchings. Also we give some lower bounds on the eccentric connectivity index of unicyclic graphs with given matching numbers.

A matching covered graph is a nontrivial connected graph in which every edge is in some perfect matching. A matching covered graph G is near-bipartite if it is nonbipartite and there is a removable double ear R such that G−R is matching covered and bipartite. In 2000, C. Little and I. Fischer characterized Pfaffian near-bipartite graphs in terms of forbidden subgraphs [3]. However, their charac...

Matching preclusion is a measure of robustness in the event of edge failure in interconnection networks. The matching preclusion number of a graph G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching, and the conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resultin...

This paper studies the classic minimum-cost perfect matching problem in complete graphs under certain stability constraints. Given some real α ≥ 1, a matching M is said to be α-stable if there does not exist any edge (x, y) / ∈ M such that α · w(x, y) < min{w(x, x′), w(y, y′)}, where x′ and y′ are the vertices matched to x and y, respectively, under M . Given some real β ≥ 1, the perfect matchi...

In this paper, we prove that there is no r-regular graph (r ≥ 2) with a unique perfect matching. Also we show that a 2r-regular graph of order n has at least ( (r−k)r−k (r−k+1)r−k−1 )n 2k-factors, where k ≤ r. We investigate graphs with a unique [a, b]-factor and among other results, we prove that a connected graph with minimum degree at least 2 and a unique [1, 2]factor with regular components...

In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erdős-Ko-Rado theorem for perfect matchings. The proof given in this paper is algebraic, we first determine the least eigenvalue of the perfect matching derangement graph and use properties...

For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is, H}$ is chosen uniformly from $M$-subsets of K}:={V \choose r}$ ($:= \{\mbox{$r$-subsets $V$}\}$). Shamir's Problem (circa 1980) asks, roughly, for what $M=M(n)$ likely to contain a perfect matching (that $n/r$ disjoint $r$-sets)? In 2008 Johan...