On eccentricity matrices of wheel graphs

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 not equivalent to 1 (mod 3). We determine the inertia of E(W-n) by obtaining the determinant and inertia of the eccentricity matrix of the fan graph. Further, we derive a formula for the inverse of E(W-n) by finding a vector w is an element of R-n and an nxn symmetric Laplacian-like matrix (L) over tilde of rank n - 1 such that E(W-n)(-1) = -1/2 (L) over tilde + 6/n - 1 ww '.