Abstract

The singular values of a matrix A are defined as the square
roots of the eigenvalues of A∗A, and the energy of A denoted
by E(A) is the sum of its singular values. The energy of
a graph G, E(G), is defined as the sum of absolute values
of the eigenvalues of its adjacency matrix. In this paper,
we prove that if A is a Hermitian matrix with the block
form A = DB D ∗ C , then E(A) ≥ 2E(D). Also, we show
that if G is a graph and H is a spanning subgraph of G
such that E(H) is an edge cut of G, then E(H) ≤ E(G),
i.e., adding any number of edges to each part of a bipartite
graph does not decrease its energy. Let G be a connected
graph of order n and size m with the adjacency matrix A.
It is wellknown that if G is a bipartite graph, then E(G) ≥
4m + n(n − 2) det(A) n2 . Here, we improve this result by
showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for
E(G) given in Oboudi (2019) [14].
