Study Of Several Classes Of Generalized Corona Graphs

Since the spectral graph has been widely applied in the field of computer science,communication network, quantum chemistry and so on, characteristic polynomials are also of great significance where the spectrum of a graph can be directly calculated by it’s characteristic polynomials. The adjacency matrix A(G) of a graph G represents the connection relationship between every two vertices in G . The characteristic polynomial of a graph’s adjacency matrix is called adjacency characteristic polynomial. Let D(G) be the diagonal matrix with diagonal entries the degrees of G. The Laplacian matrix of G is defined as L(G) = D(G) - A(G) and it’s characteristic polynomial is called Laplacian characteristic polynomial. The signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G) and it’s characteristic polynomial is called signless Laplacian characteristic polynomial. The collections of eigenvalues of those three characteristic polynomials that defined above together with their multiplicities are called the adjacency spectrum, the Laplacian spectrum and the signless Laplacian spectrum of G, respectively.Graphs with the same spectrum but not isomorphic with each other are called cospectral graphs. According to different characteristic polynomials, the cospectal graphs are so called adjacency cospectral graphs(denoted by A- cospectral graphs), Laplacian cospectral graphs(denoted by L - cospectral graphs) and signless Laplacian cospectral graphs(denoted by Q-cospectral graphs).The coronal graph is a complex graph that constructed by multiple graphs. The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G . In this paper we define two new kinds of graph operators by generalizing the coronal graph and combining with the subdivision graph: generalized subdivision coronaedge graph S(G)(?)Hi and generalized subdivision corona vertex graph S(G)(?)Hi.Then by using the block matrix, Schur complement, the coronal of a matrix and several other tools, the adjacency characteristic polynomial, Laplacian characteristic polynomial and signless Laplacian characteristic polynomial were explicitly computed. As applications, the number of spanning trees and Kirchhoff index of those newly defined generalized coronal graphs under special conditions were calculated. As for the corollaries about number of spanning trees, examples are given to verify the results. Then families of A-cospectral,L - cospectral and Q- cospectral graphs are also obtained and examples are given.In this thesis, we get the main results as follows:(1) Define a new kind of generalized coronal graph, the generalized subdivision corona edgegraph S(G)(?)Hi. Calculate and proof the adjacency, Laplacian and signless Laplaciancharacteristic polynomial of this graph.(2) Calculate and proof the Laplacian spectrum, number of spanning trees and Kirchhoff index of S(G)(?)while graph G is an r-regular graph and H1,...Hm are the same graph H. An example of calculating the number of spanning trees is given.(3) Obtain families of A - cospectral, L- cospectral and Q- cospectral generalized subdivision corona edge graphs. Examples of cospectral graphs are given.(4) Define a new kind of generalized coronal graph, the generalized subdivision coronavertex graph S(G)(?)Hi. Calculate and proof the adjacency, Laplacian and signlessLaplacian characteristic polynomial of this graph.(5) Calculate and proof the Laplacian spectrum, number of spanning trees and Kirchhoff index of S(G)(?)H while graph G is an r-regular graph and H1,...Hn are the same graph H. An example of calculating the number of spanning trees is given.(6) Obtain families of A - cospectral, L- cospectral and Q- cospectral generalized subdivision corona vertex graphs. Examples of cospectral graphs are given.