On Extremal Skew Energy Of Unicyclic Graphs With Given Pendent Vertices

Graph theory is an important branch of combinatorial mathematics. It has a wide range of important application in fields such as computer, chemical, physics. Let G be a simple graph. Tho energy of the graph E(G) is defined as the sum of the absolute value of all the eigenvalues of the adjacency matrix A(G). In recent years, with the development of graph energy, many scholars begin to pay attention to study the energy of other matrices, such as the Laplacian matrix, incidence matrix and the distance matrix. Let G be an oriented graph of graph G, the skew energy ES(G) of the oriented graphs G is defined as the sum of the norms of all the eigenvalues of the skew adjacency matrix S(G). In the study of graph energy and the skew energy, extreme energy problem with graphic parameters has been a hot research problem in graph theory.This thesis mainly studies the skew energy extremal problem of uni-cyclic graphs with the given pendent vertices. Let G(n. k) be the given pendent vertices of unicyclic graphs to remove Qnl,k. Let G(n,l,k) be a unicyclic graph class of G(n,k) with prescribed girth. The main result of this thesis are follow-ing:(1) show that -Rnl,k(3≤l<≤n-k-1) is the minimum skew energy of the oriented unicyclic graph among the unicyclic graph class of G(n,l,k) (3≤l≤n-k-1) with prescribed girth.,-Rn4,k is the minimum skew energy among the unicyclic graph class of -Rnl,k≤l≤n-k-1). Namely,-Rn4,k is the minimum skew energy of among the unicyclic graph class of G(n, k).(2)show that-Qn4,k is the minimum skew energy of the oriented unicyclic graph among the unicyclic graph class of Qnl,k (3≤l≤n-k-1).