The energy of C₄-free graphs of bounded degree

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, satisfies

$$\mid {\mu }_1\mid +\cdots +\mid {\mu }_n\mid ⩾n\text{,}$$

where ${\mu }_1\text{,}\dots \text{,}{\mu }_n$ are the eigenvalues of G.

We give some general results and state two conjectures.