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IMCS/Publications/CSJM/Issues/CSJM v.26, n.2 (77), 2018/

Degree subtraction eigenvalues and energy of graphs

Authors: H. S. Ramane, K. C. Nandeesh, G. A. Gudodagi, B. Zhou
Keywords: Degree of a vertex, degree subtraction matrix, eigenvalues, energy, first Zagreb index.

Abstract

The degree subtraction matrix $DS(G)$ of a graph $G$ is introduced, whose $(j,k)$-th entry is $d_G(v_j) - d_G(v_k)$, where $d_G(v_j)$ is the degree of a vertex $v_j$ in $G$. If $G$ is a non-regular graph, then $DS(G)$ has exactly two nonzero eigenvalues which are purely imaginary. Eigenvalues of the degree subtraction matrices of a graph and of its complement are the same. The degree subtraction energy of $G$ is defined as the sum of absolute values of eigenvalues of $DS(G)$ and we express it in terms of the first Zagreb index.

H. S. Ramane
Department of Mathematics,
Karnatak University,
Dharwad - 580003, India
E-mail:

K. C. Nandeesh
Department of Mathematics,
Karnatak University,
Dharwad - 580003, India
E-mail:

G. A. Gudodagi
Department of Mathematics,
Karnatak University,
Dharwad - 580003, India
E-mail:

B. Zhou
Department of Mathematics,
South China Normal University,
Guangzhou 510631, P. R. China
E-mail:

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