A graph inequality on the common neighbourhood

TL;DR

This paper establishes a new graph inequality involving common neighbourhood sizes, characterizes extremal graphs, and offers both a proof derived from electric network theory and a combinatorial proof inspired by Turán's theorem.

Contribution

It introduces a novel graph inequality based on common neighbourhoods and characterizes the extremal graphs achieving equality, connecting electric network theory with combinatorial methods.

Findings

Proved a new graph inequality involving common neighbourhoods.

Characterized extremal graphs that attain equality in the inequality.

Provided two proofs: one from electric network theory and one combinatorial.

Abstract

In this note we prove a graph inequality based on the sizes of the common neighbourhoods. We also characterize the extremal graphs that achieve the equality. The result was first discovered as a consequence of the classical Forster's theorem in electric networks. We also present a short combinatorial proof that was inspired by a similar inequality related to the celebrated Tur\'an's theorem.