On maximum degree (signless) Laplacian matrix of a graph
DOI:
https://doi.org/10.22199/issn.0717-6279-5434Keywords:
maximum degree matrix, maximum degree Laplacian matrix, maximum degree signless Laplacian matrixAbstract
Let G be a simple graph on n vertices and v1, v2, . . . , vn be the vertices of
G. We denote the degree of a vertex vi in G by dG(vi) = di. The maximum
degree matrix of G, denoted by M(G), is the real symmetric matrix with
its ijth entry equal to max{di, dj} if the vertices vi and vj are adjacent in
G, 0 otherwise. In analogous to the definitions of Laplacian matrix and
signless Laplacian matrix of a graph, we consider Laplacian and signless
Laplacian for the maximum degree matrix, called the maximum degree
Laplacian matrix and the maximum degree signless Laplacian matrix,
respectively. Also, we introduce maximum degree Laplacian energy and
maximum degree signless Laplacian energy of a graph. Then we determine
the maximum degree (signless) Laplacian energy of some graphs in terms
of ordinary energy, and (signless) Laplacian energy. We compute the
maximum degree (signless) Laplacian spectra of some graph compositions.
A lower and upper bound for the largest eigenvalue of the (signless) Laplacian
matrix is established and also we determine an upper bound for the second
smallest eigenvalue of maximum degree Laplacian matrix in terms of vertex
connectivity. We also determine bounds for the maximum degree (signless)
Laplacian energy in terms of first Zagreb index.
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