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Erschienen in:
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2012 | OriginalPaper | Buchkapitel

5. Bounds for the Energy of Graphs

verfasst von : Xueliang Li, Yongtang Shi, Ivan Gutman

Erschienen in: Graph Energy

Verlag: Springer New York

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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 λ1>λ2 [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:

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Vorheriges Kapitel Common Proof Methods
Nächstes Kapitel The Energy of Random Graphs
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Metadaten
Titel
Bounds for the Energy of Graphs
verfasst von
Xueliang Li
Yongtang Shi
Ivan Gutman
Copyright-Jahr
2012
Verlag
Springer New York
Buch
Graph Energy

Print ISBN: 978-1-4614-4219-6

Electronic ISBN: 978-1-4614-4220-2

Copyright-Jahr: 2012

DOI
https://doi.org/10.1007/978-1-4614-4220-2_5