Associating hub‐directed multigraphs to arrowhead matrices

Abstract

In this paper, we use the arrowhead matrices as a tool to study graph theory. More precisely, we deal with an interesting class of directed multigraphs, the hub‐directed multigraphs. We associate the arrowhead matrices with the adjacency matrices of a class of directed multigraph, and we obtain new properties of the second objects by using properties of the first ones. The hub‐directed multigraphs with potential use in applications are also defined. As main result, we show that a hub‐directed multigraph G(H) with adjacency matrix H∗ is a dominant hub‐directed multigraph if and only if H∗=CE, where C is the adjacency matrix of another directed multigraph and E is the adjacency matrix of a particular elementary dominant hub‐directed pseudo‐graph. Another decomposition of its Gram (arrowhead) matrix A=()H∗TH∗ is also given. Copyright © 2017 John Wiley & Sons, Ltd.

Determinants of Arrowhead Matrices over Finite Commutative Chain Rings

“…Arrowhead matrices have applications in various fields, e.g., wireless communications in [15], eigenvalue decompositions of some matrices in [16], the study of directed multigraphs and hub-directed multigraphs in [12], and the study of disordered quantum spins in [4]. As a generalization of [8], the enumeration of arrowhead matrices with prescribed determinant over a FCCR is investigated in the following set up.…”