Graph Energy pp 59-81

# Bounds for the Energy of Graphs

• Xueliang Li
• Yongtang Shi
• Ivan Gutman
Chapter

## Abstract

A graph G of order n and size m is called an (n, m)-graph. In what follows we assume that the graph eigenvalues are labeled in a nonincreasing manner, i.e., λ1≥λ2≥⋯≥λ n . If G is connected, then λ12 [81]. Because λ1≥|λ i |,i=2,,n, the eigenvalue λ1 is referred to as the spectral radius of G. Three well-known relations for the eigenvalues are
$$\begin{array}{rcl} & & \sum\limits_{i=1}^{n}{\lambda }_{ i} = 0\end{array}$$
(5.1)
$$\begin{array}{rcl} & & \sum\limits_{i=1}^{n}{\lambda }_{ i}^{2} = 2m\end{array}$$
(5.2)
$$\begin{array}{rcl} & & \sum\limits_{i<j}{\lambda }_{i}\,{\lambda }_{j} = -m.\end{array}$$
(5.3)
The following lemma [81] will be frequently used in the proofs:

### Keywords

Hydrocarbon Hexagonal

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