1. On two conjectures of Randić index and the largest signless Laplacian eigenvalue of graphs.
- Author
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Deng, Hanyuan, Balachandran, S., and Ayyaswamy, S.K.
- Subjects
- *
EIGENVALUES , *GRAPH theory , *LAPLACIAN matrices , *HARMONIC analysis (Mathematics) , *MATHEMATICAL proofs , *MATHEMATICAL analysis - Abstract
Abstract: The Randić index R of a graph G is defined as the sum of over all edges of G, where denotes the degree of a vertex in G. is the largest eigenvalue of the signless Laplacian matrix of G, where D is the diagonal matrix with degrees of the vertices on the main diagonal and A is the adjacency matrix of G. Hansen and Lucas [18] conjectured (1) and equality holds for and (2) with equality if and only if for and for , respectively. In this paper, we prove the conjecture (1) and obtain a result very close to the conjecture (2). Moreover, we give some results relating harmonic index and the largest eigenvalue of the adjacency matrix. [Copyright &y& Elsevier]
- Published
- 2014
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