## 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], = 2*m*.

In what follows we assume that the graph eigenvalues are labeled in a nonincreasing manner, i.e., that

If *G* is connected, then λ_{1} > λ_{2} [1]. Because λ_{1} ≥ |λ_{i}|, *i* = 2,..., *n*, the eigenvalue λ_{1} 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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