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Let G be a graph with adjacency matrix A(G), and let D(G) be the diagonal matrix of the degrees of G. For any real number α ∈ [0,1], Nikiforov defined the Aα-matrix of G as Aα (G) = αD(G) + (1 - α)A(G). The eigenvalues of the matrix Aα(G) form the Aα-spectrum of G. The Aα-spectral radius of G is the largest eigenvalue of Aα(G) denoted by ρα(G). In this paper, we propose the Aα--matrix of G as Aα-(G) = αD(G) + (α - 1)A(G), 0 ≤ α ≤ 1. Let the Aα--spectral radius of G be denoted by λα-(G), and let Sβα(G) and Sβα- (G) be the sum of the βth powers of the Aα and Aα- eigenvalues of G, respectively. We determine the Aα--spectra of some graphs and obtain some bounds of the Aα--spectral radius. Moreover, we establish a relationship between the Aα-spectral radius and Aα--spectral radius. Indeed, for α ∈ (1/2, 1), we show that λα- ≤ ρα, and we prove that if G is connected, then the equality holds if and only if G is bipartite. Employing this relation, we obtain some upper bounds of λα- (G), and we prove that the Aα--spectrum and Aα-spectrum are equal if and only if G is a bipartite connected graph. Furthermore, we generalize the relation established by S. Akbari et al. in (2010) as follows: for α ∈ [1/2, 1), if 0 < β ≤ 1 or 2 ≤ β ≤ 3, then Sβα(G) ≥ Sβα- (G), and if 1 ≤ β ≤ 2, then Sβα(G) ≤ Sβα-(G), where the equality holds if and only if G is a bipartite graph such that β ∉ {1,2,3}. [ABSTRACT FROM AUTHOR]

Let R be a ring and U (R) be the set of unit elements of R. The unit graph G(R) of R is the graph whose vertices are all the elements of R, defining distinct vertices x and y to be adjacent if and only if x + y ∈ U (R). The Laplacian spectrum of G(ℤn) was studied when n = pm, where p is a prime and m is a positive integer. Consequently, in this paper, we study the Laplacian spectrum of G(ℤn), for n = p1p2...pk, where pi are distinct primes and i = 1,2, ..., k. [ABSTRACT FROM AUTHOR]

Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and bonds by the edges of this graph. Topological graph indices have been successfully used in determining the structural properties and in predicting certain physicochemical properties of chemical compounds. Wiener index is the oldest topological index which can be used for analyzing intrinsic properties of a molecular structure in chemistry. The Wiener index of a graph G is equal to the sum of distances between all pairs of vertices of G. Recently, the entire versions of several indices have been introduced and studied due to their applications. Here we introduce the entire Wiener index of a graph. Exact values of this index for trees and some graph families are obtained, some properties and bounds for the entire Wiener index are established. Exact values of this new index for subdivision and k-subdivision graphs and some graph operations are obtained. [ABSTRACT FROM AUTHOR]