The conditional matching preclusion number of a graph with n vertices is the minimum number of edges whose deletion results in a graph without an isolated vertex that does not have a perfect matching if n is even, or an almost perfect matching if n is odd. We develop some general properties on conditional matching preclusion and then analyze the conditional matching preclusion numbers for some ...

The line element of a class solutions which match to the solution Kerr on an oblate spheroid if two functions $ F(r)$ and $H(r)$ it depends satisfy certain matching conditions is presented. non vanishing components Ricci tensor $R_{\mu\nu}$, scalar R$, second order curvature invariant $K$, eigenvalues tensor, energy density $\mu$, tangential pressure $P_{\perp}$, quantity $\mu+P_{\perp}$ are ca...

The Randić index R(G) of a graph G is the sum of the weights (d(u)d(v))− 1 2 of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on the Randić index of unicyclic graphs is given in terms of the order and given ...

Today, we will use an algebraic approach to solve the matching problem. Our goal is to derive an algebraic test for deciding if a graph G = (V, E) has a perfect matching. We may assume that the number of vertices is even since this is a necessary condition for having a perfect matching. First, we will define a few basic needed notations.

Matching is an extensively studied topic. The question we want to study here is the lower bound of a maximum matching in a graph. Earlier researchers studied the problem of the existence of a perfect matching (i.e. a matching of size n/2 with n vertices in a graph). Petersen [7] showed that a bridgeless cubic graph has a perfect matching. König [4] showed that there exists a perfect matching in...

A matching in a graph G is a set M = {e1, e2, . . . , ek} of edges such that each vertex v ∈ V (G) appears in at most one edge of M . That is, ei ∩ ej = ∅ for all i, j. The size of a matching is the number of edges that appear in the matching. A perfect matching in a graph G is a matching in which every vertex of G appears exactly once, that is, a matching of size exactly n/2. Note that a perfe...

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those inci...

We consider three restrictions on Boolean circuits: bijectivity, consistency and mul-tilinearity. Our main result is that Boolean circuits require exponential size to compute the bipartite perfect matching function when restricted to be (i) bijective or (ii) consistent and multilinear. As a consequence of the lower bound on bijec-tive circuits, we prove an exponential size lower bound for monot...

A perfect matching covering of a graph G is a set of perfect matchings of G such that every edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 (we call such a collection of perfect matchings a Berge covering of G). A cubic graph G is called a Kotzig graph if G has a 3-edge-coloring such t...