The Laplacian-energy-like Invariant,Incidence Energy And Kirchhoff Index Of Some Graphs

As an important branch of Algebraic Graph Theory, the spectral graph theory mainly concerns the relation between the combinatorial properties of a graph and the algebraic properties of matrices (such as adjacency matrix, Laplacian matrix, incidence matrix, etc.) associated with the graph. It has broad but important application in physics, quantum chemistry, information science and communication networks and so on.This thesis does some researches on Laplacian-energy-like invariant of a graph, incidence energy of a graph and Kirchhoff indices of some graphs, ob-tains some new meaningful results, and consists of the following five chapters.In Chapter one, besides introducing the background and developments of the graph energy, some fundamental concepts, terminologies and notations, we give a brief induction to main results of our work.In Chapter2, wo first discuss the bounds for LEL of graphs, and obtain a new lower bound. Then, for line graph, subdivision graph, total graph of a regular graph and the line graph of a semiregular graph, we give the bounds for LEL of them in terms of vertices and regularity (or semi-regularity), and determine the graphs which the bounds are sharp.In Chapter3, we investigate the asymptotic behavior of LEL of some special graphs. In particular, we show that the asymptotic value of the LEL of iterated line graph of a regular graph G is independent of the structure of G. For square lattices with toroidal (or cylindrical, free) boundary conditions, the hexagonal and triangular lattices with toroidal boundary condition, we prove that the growth rate of the LEL of them is only dependent on the number of vertices of them.In Chapter4, we mainly study the bounds for IE of graphs, and obtain a new lower bound. Then for total graph of a regular graph and the line graph of a semiregular graph, we establish the relation between the signless Laplacian polynomials of them and the "original graph". Based on this, we give the bounds for IE of them in terms of vertices and regularity (or semi-regularity), and determine the graphs which the bounds are sharp.In Chapter5, we consider the Kirchhoff index of some graphs. Firstly, for line graph, subdivision graph. total graph of a regular graph and the line graph of a semiregular graph, we give the bounds for kirehhoff indices of them in terms of vertices and regularity (or semi-regularity), determine the graphs which the bounds are sharp, and obtain the formulas for Kirchhoff index of them. Secondly, we establish the relationship between the Laplacian polynomials of R(G) and Q(G) and the Laplacian polynomials of G, where R(G) and Q(G) are the graph obtained by operators R and Q act on a regular graph G. Based on this, we get the formulas for Kirchhoff index of R(G) and Q(G).