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On the Sum of the Powers of Distance Signless Laplacian Eigenvalues of Graphs

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Let G be a connected graph with n vertices, m edges and having distance signless Laplacian eigenvalues ρ1≥ ρ2… ≥ ρn 0. For any real number α ≠ 0, let \({m_\alpha }\left( G \right) = \sum\nolimits_{i = 1}^n {\rho _i^\alpha } \) be the sum of αth powers of the distance signless Laplacian eigenvalues of the graph G. In this paper, we obtain various bounds for the graph invariant mα(G), which connects it with different parameters associated to the structure of the graph G. We also obtain various bounds for the quantity DEL(G), the distance signless Laplacian-energy-like invariant of the graph G. These bounds improve some previously known bounds. We also pose some extremal problems about DEL(G).

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The authors thank the referee for his useful comments and suggestions. The research of S. Pirzada is supported by SERB-DST, New Delhi, under the research project number MTR/2017/000084.

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Correspondence to S. Pirzada, Hilal A. Ganie, A. Alhevaz or M. Baghipur.

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Pirzada, S., Ganie, H.A., Alhevaz, A. et al. On the Sum of the Powers of Distance Signless Laplacian Eigenvalues of Graphs. Indian J Pure Appl Math 51, 1143–1163 (2020).

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2010 Mathematics Subject Classification