Authors: Feng, Lihua
Lu, Lu
Stevanović, Dragan 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Walk counting and Nikiforov’s problem
Journal: Open Journal of Discrete Applied Mathematics
Volume: 3
Issue: 1
First page: 11
Last page: 19
Issue Date: 10-Feb-2020
Rank: M53
DOI: 10.30538/psrp-odam2020.0024
URL: https://pisrt.org/psrpress/j/odam/2020/1/3/walk-counting-and-nikiforov-s-problem.pdf
Abstract: 
For a given graph, let wk denote the number of its walks with k vertices and let λ1 denote the spectral radius of its adjacency matrix. Nikiforov asked in [Linear Algebra Appl 418 (2006), 257–268] whether it is true in a connected bipartite graph that λr1≥ws+rws for every even s≥2 and even r≥2? We construct here several infinite sequences of connected bipartite graphs with two main eigenvalues for which the ratio ws+rλr1ws is larger than~1 for every even s,r≥2, and thus provide a negative answer to the above problem.
Keywords: Walks in a graph | spectral radius | main eigenvalues
Publisher: PSR Press
Project: Graph theory and mathematical programming with applications in chemistry and computer science 

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