The energy of C4-free graphs of bounded degree

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∣μ1∣+⋯+∣μn∣⩾n,∣μ1∣+⋯+∣μn∣⩾n,where μ1,…,μnμ1,…,μn are the eigenvalues of G.We give some general results and state two conjectures.