Abstract

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al.
proved that for every cubic graph of order n, E(G) ≥ n. Here, we improve this result by showing that if G is a connected
subcubic graph of order n ≥ 8, then E(G) ≥ 1.01n. Also, we prove that if G is a traceable subcubic graph of order n ≥ 8,
then E(G) > 1.1n. Let G be a connected cubic graph of order n ≥ 8, it is shown that E(G) > n + 2. It was proved that if
G is a connected cubic graph of order n, then E(G) ≤ 1.65n. Also, in this paper we would like to present the best lower
bound for the energy of a connected cubic graph. We introduce an infinite family of connected cubic graphs whose for each
element of order n, say G, E(G) ≥ 1.24n, and conjecture that if 6n, then minimum energy occurs just for each element of
this family. We conjecture that there exists N such that for every connected cubic graph G of order n ≥ N, E(G) ≥ 1.24n.
