On Ordering Of Complements Of Graphs With Respect To Matching Numbers

Matching numbers of the quasi-order relation is leaded to matching numbers with some im-portant applications in the energy of a graph, matching energy of a graph, Hosoya index of a graph and so on. This paper apply quasi-order of matching numbers in the matching energy to study on ordering of complements of graphs with respect to matching numbers and based on Parameter Control, we study the complements of unicyclic graphs with given order and unicyclic graphs with given order and girth.In the second chapter of this paper, We characterize some transformations on the complements of graphs that increase or decrease the matching numbers accordingly. For example, we characterize the grafting transformation of the graph with two pendent paths for the influence of matching numbers of complements of graph when these two pendent paths at one vertex, or two adjacent vertices, or two non-adjacent vertices. And we research that if G1 is the graph obtained from G by inserting t vertices into the edge uv, G2 is the graph obtained from G by joining the vertex u to an end vertex of path Pt, then G1≥G2.Further study, for e= uv is an arbitrary edge of G where dG(u)> 1 and dG(v)> 1, let G(u o v) denote the graph obtained from G by identifying the vertex u and the vertex v and then adding a pendent edge e to the new vertex, then G>G(u o v).In the third chapter of this paper, we apply some results of the second chapter of this paper to the specific graphs based on Parameter Control. We characterize on ordering of complements of unicyclic graphs with given order and unicyclic graphs with given order and girth.