On Eccentricity Matrices of Wheel Graphs

Abstract

The eccentricity matrix E(G) of a simple connected graph G is obtained from the distance matrix D(G) of G by retaining the largest distance in each row and column, and by defining the remaining entries to be zero. This paper focuses on the eccentricity matrix E(W n ) of the wheel graph W n with n vertices. By establishing a formula for the determinant of E(W n ), we show that E(W n ) is invertible if and only if n ≡ 1 (mod 3). We derive a formula for the inverse of E(W n ) by finding a vector w ∈ R n and an n × n symmetric Laplacian-like matrix L of rank n − 1 such thatFurther, we prove an analogous result for the Moore-Penrose inverse of E(W n ) for the singular case. We also determine the inertia of E(W n ).