7.2 Bounds for the Energy of Graphs
Let G be a graph possessing n vertices and m edges. We say that G is an (n, m)-graph.
For any (n, m)-graph [1], 
= 2m.
In what follows we assume that the graph eigenvalues are labeled in a nonincreasing manner, i.e., that
If G is connected, then λ1 > λ2 [1]. Because λ1 ≥ |λi|, i = 2,..., n, the eigenvalue λ1 is referred to as the spectral radius of graph G.
Some of the simplest and longest standing [8] bounds for the energy of graphs are given below.
Theorem 7.1 [11] For an (n, m)-graph G,
with equality if and only if G is either an empty graph (with m = 0, i.e., 
)or a regular graph of degree 1, i.e., 
.
Theorem 7.2 [12] For a graph G with m edges,
Equality E (G) = 
holds if and only if G consists of a complete bipartite graph K a,b, such that a· b = m, and ...
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