The energy of C4-free graphs of bounded degree

TL;DR

This paper investigates the energy of C4-free graphs with bounded degree, proving that most such graphs have energy exceeding their order, and introduces new theorems and conjectures in spectral graph theory.

Contribution

It establishes that nearly all connected C4-free graphs with maximum degree 3 have energy greater than their order, except for four specific trees, and presents new theorems and conjectures.

Findings

Most C4-free graphs with degree ≤ 3 have energy > number of vertices

Four specific trees are exceptions to the energy bound

New theorems and conjectures in spectral graph theory

Abstract

Answering some questions of Gutman, we show that, except for four specific trees, every connected graph G of order n, with no cycle of order 4 and with maximum degree at most 3, has energy greater that its order. Here, the energy of a graph is the sum of the moduli of its eigenvalues. We give more general theorems and state two conjectures.