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In this paper the unique graph with maximal index (i.e. the largest eigenvalue of the adjacency matrix) is identified among all connected bipartite graphs of order n and size n + k , under the assumption that k ≥ 0 and n ≥ k + 5 . [ABSTRACT FROM AUTHOR]

Let G be a simple graph with the adjacency matrix A ( G ) . A well-known result of Cvetković and Gutman states that removing a pendant vertex and its neighbour, does not change the nullity of the graph. We generalize this theorem and some other theorems similar to it for an arbitrary eigenvalue of a graph. Also, for an arbitrary eigenvalue λ , we use a corresponding star set to delete some subgraphs and to determine the multiplicity of λ . We use star sets to find some Parter–Wiener vertices, in trees. We state our results for real symmetric matrices (and so for weighted graphs) and as a corollary we use them for simple graphs. Furthermore, we use these methods to obtain some results for λ = 0 , ± 1 . [ABSTRACT FROM AUTHOR]

Let G = G1 ∪ G2 be the sum of two simple graphs G1, G2 having a common edge or G = G1 ∪ e1 ∪ e2 ∪ G2 be the sum of two simple disjoint graphs G1,G2 connected by two edges e1 and e2 which form a cycle C4 inside G. We give a method of computing the determinant detA(G) of the adjacency matrix of G by reducing the calculation of the determinant to certain subgraphs of G1 and G2. To show the scope and effectiveness of our method we give some examples [ABSTRACT FROM AUTHOR]

Abstract: The positive index of inertia of a graph G, denoted by , is the number of the positive eigenvalues of the adjacency matrix of G. In this paper, we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices, and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n. [Copyright &y& Elsevier]

Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: what is the structure of a connected graph G with rank 5? or equivalently, what is the structure of a connected n-vertex graph G whose adjacency matrix has nullity ? In this paper, we completely characterize connected graphs G whose adjacency matrix has rank 5. [Copyright &y& Elsevier]

Abstract: Suppose that is the graph obtained by taking the box product of a path of length and a path of length . Let be the adjacency matrix of . In 1996, Rara showed that, if , then . We extend this result to allow and to be any positive integers, and show that [Copyright &y& Elsevier]

Abstract: Let be a function. Let be the matrix defined by if for some i and j and otherwise. We describe the Jordan canonical form of the matrix in terms of the directed graph for which is the adjacency matrix. We discuss several examples including a connection with the Collatz conjecture. [Copyright &y& Elsevier]

Abstract: We consider simple graphs and their adjacency matrices. In , Rara (1996) gives methods of reducing graphs which simplify the procedure of computing the determinant of their adjacency matrices. We continue this subject matter and give a general method of reducing graphs. By the use of this method we define a formula for computing the determinant of any planar grid and in particular settle the problem of their singularity. [Copyright &y& Elsevier]

Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: What is the structure of a connected graph with rank 4? This question has not yet been fully answered in the literature, and only some partial results are known. In this paper we resolve this question by completely characterizing graphs G whose adjacency matrix has rank 4. [Copyright &y& Elsevier]

Abstract: The nullity of a graph is defined as the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph. We investigate a class of graphs with pendant trees, and express the nullity of such graph in terms of that of its subgraphs. As an application of our results, we characterize unicyclic graphs with a given nullity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs of a fixed order and a given vertex (edge) independence number or vertex (edge) cover number, and get some bounds for the vertex (edge) independence number, vertex (edge) cover number of a graph in terms of the least eigenvalue of the graph. [Copyright &y& Elsevier]

Abstract: A semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency/Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence. [Copyright &y& Elsevier]

Abstract: We determine all nested split graphs (i.e. graphs having no induced subgraphs equal to or ) having the second largest eigenvalue less than 1 and give some data and observations regarding obtained results. [Copyright &y& Elsevier]

Abstract: Let G be a simple graph. Let and denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G, respectively. Let denote the largest vertex degree. If G has just one cycle, thenThe equality holds if and only if . AndThe equality holds if and only if , n is even. [Copyright &y& Elsevier]