Euclidean Distance Matrices and Inverses of Matrices of Graphs

Abstract

In this thesis, we study the concepts of Euclidean and circum-Euclidean distance matrices, newlineand the problem of finding elegant formulae for the inverses and the Moore-Penrose newlineinverses of certain matrices including the matrices associated with several graphs. newlineMotivated by the inverse formula of the distance matrix of a tree and the Moore-Penrose newlineinverse of a circum-Euclidean distance matrix (CEDM), in the first part of this thesis, newlinewe study a general real square matrix and#119872; whose Moore-Penrose inverse can be expressed newlineas the sum of a Laplacian-like matrix and#119871; and a rank one matrix. In particular, for a newlinesymmetric hollow matrix and#119872;, under an assumption, we show that and#119872; is an Euclidean newlinedistance matrix if and only if and#119871; is a positive semidefinite matrix. Based on this, we newlineobtain a new characterization of CEDMs involving their Moore-Penrose inverses. As an newlineapplication, we show that the distance matrices of weighted trees, block graphs and oddcycle- newlineclique graphs are CEDMs. Further, we establish an interlacing property between newlinethe eigenvalues of an Euclidean distance matrix and#119872; (including the singular case) and its newlineassociated Laplacian-like matrix and#119871;, which generalizes the interlacing property proved newlinefor the distance matrices of trees. newlineIn the second part of this thesis, we present necessary and sufficient conditions for a newlinegeneral real symmetric matrix and, in particular, for certain real bordered matrices to newlinehave their Moore-Penrose inverses as the sum of a Laplacian-like matrix and a rank newlineone matrix. Based on these characterizations, we obtain explicit formulae for the newlineMoore-Penrose inverses of some bordered matrices. As a consequence, we recover newlinethe Moore-Penrose inverse formulae proved for the adjacency and distance matrices of newlinewheel graphs. Specializing the above necessary and sufficient conditions to the distance newlinematrix and#119863;(and#119867;and#119899;) of a helm graph and#119867;and#119899;, we provide a short proof of the inverse formula newlinegiven for the non-singular and#119863;(and#119867;and#119899;) and derive a Graham and Lovász type formula for the newlineMoore-Penrose inverse of the singular