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For 0 ≤ α ≤ 1, Nikiforov proposed to study the spectral properties of the family of matrices Aα(G) = αD(G)+(1 - α)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The α-spectral radius of G is the largest eigenvalue of Aα(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the α-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the α-spectral radius for unicyclic graphs G with maximum degree ∆ ≥ 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest α-spectral radius among trees, and the unique tree with the largest α-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the α-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively. [ABSTRACT FROM AUTHOR]

Let G be a g√aph on n ve√tices and G its complement. In this pape√, we p√ove a No√dhaus-Gaddum type inequality to the second la√gest eigenvalue of a g√aph G, λ2(G), λ2(G) + λ2(G) ≤ -1 + √ n2/2 - n + 1; when G is a g√aph with gi√th at least 5. Also, we show that the bound above is tight. Besides, we p√ove that this √esult holds fo√ some classes of connected g√aphs such as t√ees, k-cyclic, √egula√ bipa√tite and complete multipa√tite g√aphs. Based on these facts, we conjectu√e that ou√ √esult holds to any g√aph. [ABSTRACT FROM AUTHOR]