On distance matrices of helm graphs obtained from wheel graphs with an even number of vertices

TL;DR

This paper investigates the distance matrix of helm graphs derived from wheel graphs with an even number of vertices, providing explicit formulas for its determinant, inverse, and eigenvalue interlacing properties.

Contribution

It derives a closed-form expression for the determinant of the distance matrix and characterizes its inverse and eigenvalue interlacing for helm graphs with even vertices.

Findings

Determinant of the distance matrix is 3(n-1)2^{n-1}.

Explicit inverse matrix of the distance matrix is obtained.

Eigenvalues of the Laplacian matrix interlace with those of the distance matrix.

Abstract

Let $n \geq 4$. The helm graph $H_n$ on $2n-1$ vertices is obtained from the wheel graph $W_n$ by adjoining a pendant edge to each vertex of the outer cycle of $W_n$. Suppose $n$ is even. Let $D := [d_{ij}]$ be the distance matrix of $H_n$. In this paper, we first show that $\det(D) = 3(n-1)2^{n-1}.$ Next, we find a matrix $\L$ and a vector $u$ such that \[D^{-1} = -\frac{1}{2}\L+\frac{4}{3(n-1)}uu'.\] We also prove an interlacing property between the eigenvalues of $\L$ and $D$.