Kirchhoff index of composite graphs

AbstractLet G1+G2, G1∘G2 and G1{G2} be the join, corona and cluster of graphs G1 and G2, respectively. In this paper, Kirchhoff index formulae of these composite graphs are given

The Kirchhoff Index of Hypercubes and Related Complex Networks

The resistance distance between any two vertices of G is defined as the network effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all the pairs of vertices in G. We firstly provided an exact formula for the Kirchhoff index of the hypercubes networks Qn by utilizing spectral graph theory. Moreover, we obtained the relationship of Kirchhoff index between hypercubes networks Qn and its three variant networks l(Qn), s(Qn), t(Qn) by deducing the characteristic polynomial of the Laplacian matrix related networks. Finally, the special formulae for the Kirchhoff indexes of l(Qn), s(Qn), and t(Qn) were proposed, respectively

CHUNG-YAU INVARIANTS AND RANDOM WALK ON GRAPHS

The Chung-Yau graph invariants were originated from Chung-Yau’s work on discrete Green’s function. They are useful to derive explicit formulas and estimates for hitting times of random walks on discrete graphs. In this thesis, we study properties of Chung-Yau invariants and apply them to study some questions: (1) The relationship of Chung-Yau invariants to classical graph invariants; (2) The change of hitting times under natural graph operations; (3) Properties of graphs with symmetric hitting times; (4) Random walks on weighted graphs with different weight schemes

Laplacian energy of graphs and digraphs.

Spectral graph theory (Algebraic graph theory) which emerged in 1950s and 1960s is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues and eigenvectors of matrices associated to the graph. The major source of research in spectral graph theory has been the study of relationship between the structural and spectral properties of graphs. Another source has research in quantum chemistry. Just as astronomers study stellar spectra to determine the make-up of distant stars, one of the main goals in spectral graph theory is to deduce the principal properties and structure of a graph from its graph spectrum (or from a short list of easily computable invariants). The spectral approach for general graphs is a step in this direction.Digital copy of Thesis.University of Kashmir