Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The cardinality of any smallest maximal matching in $G$ is the saturation number of $G$ and is denoted by $s(G)$. In this paper we study the saturation numbe...

The purpose of the matching interdiction problem in a weighted graph is to find two vertices such that the weight of the maximum matching in the graph without these vertices is minimized. An approximate solution for this problem has been presented. In this paper, we consider dendrimers as graphs such that the weights of edges are the bond lengths. We obtain the maximum matching in some types of...

We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...

In an instance G = (A∪B,E) of the stable marriage problem with strict and possibly incomplete preference lists, a matchingM is popular if there is no matchingM ′ where the vertices that prefer M ′ to M outnumber those that prefer M to M ′. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a mi...

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

altan derivatives of polycyclic conjugated hydrocarbons were recently introduced and studied in theoretical organic chemistry. we now provide a generalization of the altan concept, applicable to any graph. several earlier noticed topological properties of altan derivatives of polycyclic conjugated hydrocarbons are shown to be the properties of all altan derivatives of all graphs. among these ar...

we characterize the rational subsets of a finite group and discuss the relations to integral cayley graphs.

We consider four weighted matching-type problems: the bipartite graph and general graph versions of matching and f -factors. The linear program duals for these problems are shown to be weights of certain subgraphs. Specifically the so-called y duals are the weights of certain maximum matchings or f -factors; z duals (used for general graphs) are the weights of certain 2-factors or 2f -factors. ...

‎in this paper‎, ‎we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is cohen-macaulay‎. ‎it is proved that if there exists a cover of an $r$-partite cohen-macaulay graph by disjoint cliques of size $r$‎, ‎then such a cover is unique‎.

the purpose of the matching interdiction problem in a weighted graph is to findtwo vertices such that the weight of the maximum matching in the graph withoutthese vertices is minimized. an approximate solution for this problem has beenpresented. in this paper, we consider dendrimers as graphs such that the weightsof edges are the bond lengths. we obtain the maximum matching in some types ofdend...