Two upper bounds on the A_α-spectral radius of a connected graph

Document Type : Original paper

Author

Department of Mathematics, Hazratbal

Abstract

If A(G) and D(G) are respectively the adjacency matrix and the diagonal matrix of vertex degrees of a connected graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G)=α D(G)+(1α) A(G), where 0α1. The Aα (or generalized) spectral radius λ(Aα(G)) (or simply λα) is the largest eigenvalue of  Aα(G). In this paper, we show that
λαα Δ+(1α)2m(11ω),
where m, Δ and  ω=ω(G) are respectively the size, the largest degree and the clique number of G. Further, if G has order n, then we show that
λα12max1in[αdi+α2di2+4mi(1α)[α+(1α)mj]],
where di and mi are respectively the degree and the average 2-degree of the vertex vi.

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