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In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ (G) = D ¯ (G) + A ̃ (G) , where D ¯ (G) is the diagonal matrix of A B C -degrees and A ̃ (G) is the A B C -matrix of G. The eigenvalues of the matrix Q ̃ (G) are the signless Laplacian A B C -eigenvalues of G. We give some basic properties of the matrix Q ̃ (G) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ (G) − t r (Q ̃ (G)) n I , called the signless Laplacian A B C -energy of G. We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy. [ABSTRACT FROM AUTHOR]

The A γ matrix of a graph G is determined by A γ (G) = (1 − γ) A (G) + γ D (G) , where 0 ≤ γ ≤ 1 , A (G) and D (G) are the adjacency and the diagonal matrices of node degrees, respectively. In this case, the A γ matrix brings together the spectral theories of the adjacency, the Laplacian, and the signless Laplacian matrices, and many more γ adjacency-type matrices. In this paper, we obtain the A γ eigenvalues of zero divisor graphs of the integer modulo rings and the von Neumann rings. These results generalize the earlier published spectral theories of the adjacency, the Laplacian and the signless Laplacian matrices of zero divisor graphs. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A (G) and D (G) and defined the A α -matrix for every real α ∈ [ 0 , 1 ] as: A α (G) = α D (G) + (1 − α) A (G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the A α -matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established. [ABSTRACT FROM AUTHOR]

Let A (G) be the adjacent matrix and D (G) the diagonal matrix of the degrees of a graph G, respectively. For 0 ≤ α ≤ 1 , the A α -matrix is the general adjacency and signless Laplacian spectral matrix having the form of A α (G) = α D (G) + (1 − α) A (G) . Clearly, A 0 (G) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A α -spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results. [ABSTRACT FROM AUTHOR]

In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ , with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ [ F ] by one of Γ \ F . A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ. [ABSTRACT FROM AUTHOR]