Sum of squares of degrees in a graph

TL;DR

This paper investigates the maximum sum of squared degrees in simple graphs with given vertices and edges, identifying which special graphs achieve this maximum and analyzing related density questions.

Contribution

It determines which of the two known extremal graphs maximizes the sum of squared degrees for each (v,e) pair and characterizes all graphs attaining the maximum.

Findings

Identifies when the extqs and extqc graphs have the same sum of squared degrees.

Classifies all graphs that achieve the maximum sum of squared degrees.

Examines density questions related to the extremal graphs.

Abstract

Let $\G(v,e)$ be the set of all simple graphs with $v$ vertices and $e$ edges and let $P_2(G)=\sum d_i^2$ denote the sum of the squares of the degrees, $d_1, >..., d_v$, of the vertices of $G$. It is known that the maximum value of $P_2(G)$ for $G \in \G(v,e)$ occurs at one or both of two special graphs in $\G(v,e)$--the \qs graph or the \qc graph. For each pair $(v,e)$, we determine which of these two graphs has the larger value of $P_2(G)$. We also determine all pairs $(v,e)$ for which the values of $P_2(G)$ are the same for the \qs and the \qc graph. In addition to the \qs and \qc graphs, we find all other graphs in $\G(v,e)$ for which the maximum value of $P_2(G)$ is attained. Density questions posed by previous authors are examined.