On maximum degree (signless) Laplacian matrix of a graph

• Rangarajan, R. ; Raghu, V. D. ^[1] ; Rakshith , B. R.
  1. [1] University of Mysore.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 41, Nº. 6, 2022, págs. 1333-1352
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-5434
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• Resumen

  • Let G be a simple graph on n vertices and v1, v2, . . . , vn be the vertices ofG. We denote the degree of a vertex vi in G by dG(vi) = di. The maximumdegree matrix of G, denoted by M(G), is the real symmetric matrix withits ijth entry equal to max{di, dj} if the vertices vi and vj are adjacent inG, 0 otherwise. In analogous to the definitions of Laplacian matrix andsignless Laplacian matrix of a graph, we consider Laplacian and signlessLaplacian for the maximum degree matrix, called the maximum degreeLaplacian matrix and the maximum degree signless Laplacian matrix,respectively. Also, we introduce maximum degree Laplacian energy andmaximum degree signless Laplacian energy of a graph. Then we determinethe maximum degree (signless) Laplacian energy of some graphs in termsof ordinary energy, and (signless) Laplacian energy. We compute themaximum degree (signless) Laplacian spectra of some graph compositions.A lower and upper bound for the largest eigenvalue of the (signless) Laplacianmatrix is established and also we determine an upper bound for the secondsmallest eigenvalue of maximum degree Laplacian matrix in terms of vertexconnectivity. We also determine bounds for the maximum degree (signless)Laplacian energy in terms of first Zagreb index.

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