Abstract

The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In Akbari and Hosseinzadeh (2020) [3] it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree δ(G) whose adjacency matrix is nonsingular, E(G) ≥ Δ(G) + δ(G) and the equality holds if and only if G is a complete graph. Let G be a connected graph with the edge set E(G). In this paper, first we show that E(L(G)) ≥E(G) + Δ(G) − 5, where L(G) denotes the line graph of G. Next, using this result, we prove the validity of the conjecture for the line of each connected graph of order at least 7.
