Matching Of Unicyclic Graphs And Estrada Index

Spectral graph theory is an important branch of algebraic graph theory which mainly deal with the relation between the structural property and the spectral property of graphs by using the spectrum of adjacency matrix and Laplacian matrix of graphs.The Estrada index of a graph G is defined as EE(G)=Σin=1eλi(G)which is an important spectral parameter of graphs, where λ1(G).λ2(G),...,λn(G) are the eigenvalues of its adjacency matrix. The Estrada index derived from the field of biochemistry. The Estrada index is an index which was introduced by Ernesto Estrada in2000to measure the degree of folding of a protein. Subsequently, The Estrada index received much attention since it was applied more and more widely in biochemistry, information science, complex network. Investigators try to characterize the structure of property of graphs by creating the relation between the Estrada index and the structural parameter of graphs. As we all know, the matching number of graph is an important structural parameter, which has a wide range of applications in many ways. This dissertation discusses the relationship between the the matching number of unicyclic graphs and Estrada index, characterizes the maximizing graph and minimizing graph among all unicyclic graphs with perfect matching, and the maximizing graph among all unicyclic graph with given small matching number.The thesis is organized as follows. In Chapter one, we introduce the background of the Estrada index of graphs, some concepts and notations which will be used in following Chap-ter. Finally, we introduce the problems, its development and our main results. In Chapter two, we mainly investigate the Estrada index of unicyclic graph with perfect matching, char-acterize the graphs whose Estrada index attain maximum and minimum among all unicyclic graphs with perfect matching, respectively. In the last Chapter, we discuss the unicyclic graph with given small matching number, characterize the maximizing graphs in unicyclic graph with matching number2and in unicyclic graph with matching number3, respectively.