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Sharp Upper Bounds on the Signless Laplacian Spectral Radius of Strongly Connected Digraphs

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Xi W., Wang L.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 4

977-988

Let G = (V (G),E(G)) be a simple strongly connected digraph and q(G) be the signless Laplacian spectral radius of G. For any vertex vi ∈ V (G), let d+i denote the outdegree of vi, m+i denote the average 2-outdegree of vi, and N+i denote the set of out-neighbors of vi. In this paper, we prove that: (1) (1) q(G) = d+1 +d+2 , (d+1 ≠ d+2) if and only if G is a star digraph [...] ,where d+1, d+2 are the maximum and the second maximum outdegree, respectively [...] is the digraph on n vertices obtained from a star graph K1,n−1 by replacing each edge with a pair of oppositely directed arcs). (2) [...] with equality if and only if G is a regular digraph. (3) [...] Moreover, the equality holds if and only if G is a regular digraph or a bipartite semiregular digraph. (4) [...] . If the equality holds, then G is a regular digraph or G ∈Ω, where is a class of digraphs defined in this paper.

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1 Discussiones Mathematicae Graph Theory

1 Wang L.

1 Xi W.

1 2016

Wyniki wyszukiwania

Sharp Upper Bounds on the Signless Laplacian Spectral Radius of Strongly Connected Digraphs