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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]

For a (simple) graph H and non-negative integers c0, c1, . . ., cd (cd ≠ 0), p(H) = Pd ...k=0 ck · Hk is the lexicographic polynomial in H of degree d, where the sum of two graphs is their join and ck · Hk is the join of ck copies of Hk. The graph Hk is the kth power of H with respect to the lexicographic product (H0 = K1). The spectrum (if H is connected and regular) and the Laplacian spectrum (in general case) of p(H) are determined in terms of the spectrum of H and ck's. Constructions of infinite families of cospectral or integral graphs are announced. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A(G), and let D(G) be the diagonal matrix of the degrees of G: The signless Laplacian Q(G) of G is defined as Q(G) := A(G) +D(G). Cvetković called the study of the adjacency matrix the A-spectral theory, and the study of the signless Laplacian-the Q-spectral theory. To track the gradual change of A(G) into Q(G), in this paper it is suggested to study the convex linear combinations Aα(G) of A(G) and D(G) defined byAα (G) := αD(G) + (1 - α )A(G) , 0 ≤ α ≤ 1. This study sheds new light on A(G) and Q(G), and yields, in particular, a novel spectral Turán theorem. A number of open problems are discussed. [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]

A graph is integral if the spectrum (of its adjacency matrix) consists entirely of integers. The problem of determining all non-regular bipartite integral graphs with maximum degree four which do not have ±1 as eigenvalues was posed in K.T. Baliéska, S.K. Simić, K.T. Zwierzyński: Which nonregular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues? Discrete Math., 286 (2004), 15-25. Here we revisit this problem, and provide its complete solution using mostly the theoretical arguments. [ABSTRACT FROM AUTHOR]

Let L (resp. L+) be the set of connected graphs with least adjacency eigenvalue at least -2 (resp. larger than -2). The nullity of a graph G, denoted by η(G), is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. In this paper, we give the nullity set of L+ and an upper bound on the nullity of exceptional graphs. An expression for the nullity of generalized line graphs is given. For G ∈ L, if η(G) is sufficiently large, then G is a proper generalized line graph (G is not a line graph). [ABSTRACT FROM AUTHOR]

This part of our work further extends our project of building a new spectral theory of graphs (based on the signless Laplacian) by some results on graph angles, by several comments and by a short survey of recent results. [ABSTRACT FROM AUTHOR]

A generalized Bethe tree is a rooted tree for which the vertices in each level having equal degree. Let Bk be a generalized BETHE tree of k level, and let Tr be a connected transitive graph on r vertices. Then we obtain a graph BkoTr from r copies of Bk and Tr by appending r roots to the vertices of Tr respectively. In this paper, we give a simple way to characterize the eigenvalues of the adjacency matrix A(BkoTr) and the Laplacian matrix L(BkoTr) of BkoTr by means of symmetric tridiagonal matrices of order k. We also present some structure properties of the Perron vectors of A(BkoTr) and the Fiedler vectors of L(BkoTr). In addition, we obtain some results on transitive graphs. [ABSTRACT FROM AUTHOR]