On the α -index of graphs with pendent paths
By Vladimir Nikiforov, Oscar Rojo. Linear Algebra and its Applications, 2017
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Abstract Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G. For every real α ∈ [ 0 , 1 ] , write A α ( G ) for the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . This paper presents some extremal results about the spectral radius ρ α ( G ) of A α ( G ) that generalize previous results about ρ 0 ( G ) and ρ 1 / 2 ( G ) . In particular, write B p , q , r be the graph obtained from a complete graph K p by deleting an edge and attaching paths P q and P r to its ends. It is shown that if α ∈ [ 0 , 1 ) and G is a graph of order n and diameter at least k, then ρ α ( G ) ≤ ρ α ( B n − k + 2 , ⌊ k / 2 ⌋ , ⌈ k / 2 ⌉ ) , with equality holding if and only if G = B n − k + 2 , ⌊ k / 2 ⌋ , ⌈ k / 2 ⌉ . This result generalizes results of Hansen and Stevanovic [5] , and Liu and Lu [7] .
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/f177bf29c78ad3395bc98df5eaad45f1a5fa4d53
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Citations in the corpus
- A note on the $A_{\alpha}$-spectral radius of graphs (2018, background): “The first lemma is due to Nikiforov and Rojo [Lemma 6, [9]], it turns out to be very useful in the paper.”