On Squared Distance Matrix of Complete Multipartite Graphs

TL;DR

This paper analyzes the squared distance matrix of complete multipartite graphs, deriving its spectral properties, energy, and extremal configurations, revealing how these metrics vary among different graph structures.

Contribution

It provides explicit formulas for the squared distance energy and inertia of complete multipartite graphs, and identifies extremal graphs maximizing or minimizing spectral radius and energy.

Findings

Squared distance energy equals 8(n - t) for certain graphs.

Bounds on energy depend on the number of singleton parts.

Complete split graphs maximize, Turán graphs minimize spectral radius and energy.

Abstract

Let $G = K_{n_1,n_2,\cdots,n_t}$ be a complete $t$-partite graph on $n=\sum_{i=1}^t n_i$ vertices. The distance between vertices $i$ and $j$ in $G$, denoted by $d_{ij}$ is defined to be the length of the shortest path between $i$ and $j$. The squared distance matrix $\Delta(G)$ of $G$ is the $n\times n$ matrix with $(i,j)^{th}$ entry equal to $0$ if $i = j$ and equal to $d_{ij}^2$ if $i \neq j$. We define the squared distance energy $E_{\Delta}(G)$ of $G$ to be the sum of the absolute values of its eigenvalues. We determine the inertia of $\Delta(G)$ and compute the squared distance energy $E_{\Delta}(G)$. More precisely, we prove that if $n_i \geq 2$ for $1\leq i \leq t$, then $ E_{\Delta}(G)=8(n-t)$ and if $ h= |\{i : n_i=1\}|\geq 1$, then $$ 8(n-t)+2(h-1) \leq E_{\Delta}(G) < 8(n-t)+2h.$$ Furthermore, we show that for a fixed value of $n$ and $t$, both the spectral radius of the squared distance matrix and the squared distance energy of complete $t$-partite graphs on $n$ vertices are maximal for complete split graph $S_{n,t}$ and minimal for Tur{\'a}n graph $T_{n,t}$.