Generalization for Laplacian energy |
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Authors: | Jian-ping Liu Bo-lian Liu |
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Institution: | [1]School of Mathematics Sciences, South China Normal University, Guangzhou 510631, China [2]College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China |
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Abstract: | Let G be a simple graph with n vertices and m edges. Let λ
1, λ
2, ..., λ
n
, be the adjacency spectrum of G, and let μ
1, μ
2, ..., μ
n
be the Laplacian spectrum of G. The energy of G is $
E\left( G \right) = \sum\limits_{i = 1}^n {\left| {\lambda _i } \right|}
$
E\left( G \right) = \sum\limits_{i = 1}^n {\left| {\lambda _i } \right|}
, while the Laplacian energy of G is defined as $
LE\left( G \right) = \sum\limits_{i = 1}^n {\left| {\mu _i - \frac{{2m}}
{n}} \right|}
$
LE\left( G \right) = \sum\limits_{i = 1}^n {\left| {\mu _i - \frac{{2m}}
{n}} \right|}
. Let γ
1, γ
2, ..., γ
n
be the eigenvalues of Hermite matrix A. The energy of Hermite matrix as $
HE\left( A \right) = \sum\limits_{i = 1}^n {\left| {\gamma _i - \frac{{tr\left( A \right)}}
{n}} \right|}
$
HE\left( A \right) = \sum\limits_{i = 1}^n {\left| {\gamma _i - \frac{{tr\left( A \right)}}
{n}} \right|}
is defined and investigated in this paper. It is a natural generalization of E(G) and LE(G). Thus all properties about energy in unity can be handled by HE(A). |
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Keywords: | energy (of matrix) Hermite matrix eigenvalue |
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