DISCRETE INVERSE CONDUCTIVITY PROBLEMS ON NETWORKS

The purpose of this dissertation is to present a mathematical model of network tomography through spectral graph theory analysis. In this regard, we explore the properties of harmonic functions and eigensystems of Laplacians for weighted graphs (networks) with and without boundary. We prove the solvability of the Dirichlet and Neumann boundary value problems. We also prove the global uniqueness of the inverse conductivity problem on a network under a suitable monotonicity condition. As a physical interpretation to the discrete inverse conductivity problem, we define a variant of the chip-firing game (a discrete balancing process) in which chips are added to the game from the boundary nodes and removed from the game if they are fired into the boundary of the graph. We find a bound on the length of the game, and examine the relations between set of spanning weighted forest rooted in the boundary of the graph and the set of critical configurations of the chips

On the distribution of eigenvalues of graphs

Let G be a simple undirected graph with n greater than or equal to 2 vertices and let alpha(0)(G) greater than or equal to ..., alpha(n-1)(G) be the eigenvalues of the adjacency matrix of G, It is shown by Cao and Yuen (1995) that if alpha(1)(G) = - 1 then G is a complete graph, and therefore alpha(0)(G) = n - 1 and alpha(i)(G) = -1 for 1 less than or equal to i less than or equal to n - 1. We obtain similar results for graphs whose complement is bipartite. We show in particular, that if the complement of G is bipartite and there exists an integer k such that 1 less than or equal to k < (n - 1)/2 and alpha(k)(G) = -1 then alpha(i)(G) = -1 for k less than or equal to i less than or equal to n - k + 1. We also compare and discuss the relation between some properties of the Laplacian and the adjacency spectra of graphs. (C) 1999 Elsevier Science B.V. All rights reserved