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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]

Let S = (G, σ) be a signed graph, where σ is the sign function on the edges of the underlying graph G. It is widely recognized that the adjacency spectrum alone cannot uniquely determine a signed graph. Therefore, it is of great interest to identify whether there exist any cospectral, non-isomorphic signed graphs within a specific class of signed graphs. In a significant contribution, Godsil et al. demonstrated that two components of G1 × G2, where both G1 and G2 are connected bipartite graphs, are cospectral if and only if at the minimum one of G1 and G2 is balanced. In this paper, we first generalize Godsil’s result for two connected signed bipartite graphs S1 and S2. Furthermore, we will define partitioned tensor product of two signed bipartite graphs, which will enable us to generate multiple pairs of cospectral non-isomorphic signed bipartite graphs. [ABSTRACT FROM AUTHOR]

Let A(G) be the adjacency matrix of a graph G. Let s(vi) denote the row entries of A(G) corresponding to the vertex vi of G. The Hamming distance between the strings s(ui) and s(vi) is the number of positions in which their elements differ. The sum of Hamming distance between all the pairs of vertices is the Hamming index of a graph. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of various products of complete bipartite and complete graph. We also compute the Hamming index generated by the adjacency matrix of these graph products. [ABSTRACT FROM AUTHOR]

A vertex u ∈ V1 in a bipartite graph G(V1∪V2;E) is redundant if all the vertices of V2 that are adjacent to u are also adjacent to a vertex w (≠= u) in V1. In other words, NG(u) ⊆ NG(w). Such vertices increase the cost of the network (when it is a communication network) or increase the unnecessary membership of the network (when it is a social network). An ideal cost effective network is the one where there is no redundant vertex. In this article, we model the above type of networks using graphs and call them antichain graphs. We characterize such graphs and study their properties. We show that if G and H are antichain graphs then so is their cartesian product G x H. We design few more methods to construct new antichain graphs from the existing ones. We also present generating procedures, which generate some regular and biregular antichain graphs. We define a critical edge with reference to the antichain property. We also characterize the critical edge. [ABSTRACT FROM AUTHOR]

In this research paper we begin with study of family of n dimensional hyper cube graphs and establish some properties related to their distance, spectra, and multiplicities and associated eigen vectors and extend to bipartite double graphs[11]. In a more involved way since no complete characterization was available with experiential results in several inter connection networks on this spectra our work will add an element to existing theory. [ABSTRACT FROM AUTHOR]

The rank of a digraph is defined as the rank of its adjacency matrix. In this paper we show how to find all digraphs of rank k from the reduced bipartite graphs of rank 2 k. In particular, since the bipartite graphs of rank 2 , 4 and 6 are known, then we characterize digraphs of rank 1 , 2 and 3. The results are based on rank-preserving operations on digraphs such as splitting and fusion of vertices or twin-deletion and vertex-multiplication. [ABSTRACT FROM AUTHOR]

The goal of this contribution is to reveal the analytical framework and synthesis guidelines for frontal-attacking targets (FRATs) and flankattacking targets (FLATs) from the point of view of a graph as a mathematical object. The final outcome of this study are three graph models that in many ways describe the shooting targets under consideration. The first graph model characterizes the structure of connections between the vertices using an undirected graph. The model showed that the complexity of silhouettes leads to an increase of the path in the graph and growth of the complexity of its internal structure. The second graph model allows the analysis of the connectivity of the graph vertices. In this case, a bipartite graph is used. As a result, the reviewed FRATs and FLATs are described by the same graph. The second model showed its indifference to the types of the used graphic primitives (GPs). The third graph model was developed for the analysis of the common borders of the neighboring GPs and it uses a bipartite graph. It is also indifferent to the types of the used GPs, but it takes into account the length of the common borders. The third model describes FRATs/FLATs groups in the same way. When using I-III models, one can design GPs and carry out the synthesis of new targets. A full group of flank-attacking targets consisting of five silhouettes and their GPs is offered. [ABSTRACT FROM AUTHOR]

We consider only connected bipartite graphs G with a unique perfect matching M . Let G w be the weighted graph obtained from G by giving weights to its edges using the positive weight function w : E ( G ) → ( 0 , ∞ ) such that w ( e ) = 1 for each e ∈ M . An unweighted graph G may be viewed as a weighted graph with the weight function w ≡ 1 . A weighted graph G w is nonsingular if its adjacency matrix A ( G w ) is nonsingular. The inverse of a nonsingular weighted graph G w is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A ( G w ) via a diagonal matrix of ±1s. By G / M we denote the graph obtained from G by contracting each edge of M to a single vertex. It is known that if G / M is bipartite, then G w is invertible for each weight function w. In this article, we consider the converse of this result and show that if G w is invertible for each w, then the graph G / M is bipartite. We also supply exact conditions under which G w is invertible for one w will force that the graph G / M is bipartite. [ABSTRACT FROM AUTHOR]

We consider only simple graphs. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite. On the class H g of such graphs G the equivalence of the following statements are known. i) The reciprocal of the spectral radius of the adjacency matrix A ( G ) is the least positive eigenvalue of the adjacency matrix. ii) The graph is isomorphic to its inverse, where the inverse of a graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A ( G ) via a diagonal matrix of ±1s. iii) The graph has the reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A ( G ) is also an eigenvalue of A ( G ) . iv) The graph has the strong reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A ( G ) is also an eigenvalue of A ( G ) and they both have the same multiplicities. v) The graph is a corona graph, that is, it is obtained by taking a bipartite graph and then by inserting a new adjacent vertex of degree one at each vertex. It is known that the statements iii) and iv) are not equivalent, if we allow nonbipartite graphs. However, the equivalence of these two is not yet known for the class of bipartite graphs with unique perfect matchings, where we relax the condition of bipartiteness of the contracted graph. In this article, we establish the equivalence of the first four statements for a class larger than H g . We then supply a constructive structural description of those graphs. [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 . For every real α ∈ [ 0 , 1 ] , write A α ( G ) for the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let α 0 ( G ) be the smallest α for which A α ( G ) is positive semidefinite. It is known that α 0 ( G ) ≤ 1 / 2 . The main results of this paper are: (1) if G is d -regular then α 0 = − λ min ( A ( G ) ) d − λ min ( A ( G ) ) , where λ min ( A ( G ) ) is the smallest eigenvalue of A ( G ) ; (2) G contains a bipartite component if and only if α 0 ( G ) = 1 / 2 ; (3) if G is r -colorable, then α 0 ( G ) ≥ 1 / r . [ABSTRACT FROM AUTHOR]

Given a graph G . The fewest number of vertices whose deletion yields a bipartite graph from G was defined by S. Fallat and Yi-Zheng Fan to be the vertex bipartiteness of G and it is denoted by υ b ( G ) . We consider the set Σ k ( n ) defined by { G = ( V ( G ) , E ( G ) ) : G connected , | V ( G ) | = n and υ b ( G ) ≤ k } . In this work we identify the graph in Σ k ( n ) with maximum spectral radius and maximum signless Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

With a nonzero rectangular ( 0 , 1 ) -matrix A we associate an undirected graph G A that corresponds to the linear transformation X ↦ A X T A . We use the generalized singular-value decomopsition of incidence matrices to count the number of connected components and the number of bipartite connected components of G A . We show that G A is connected if and only if G A has no bipartite connected component or isolated vertex. We also show that if A has no row or column of 0's, then the number of connected components is 1 2 s ( s + 1 ) and the number of bipartite connected components is 1 2 s ( s − 1 ) , where s is the number of chainable components of A , that is, the number of connected components in the undirected graph with adjacency matrix ( O A A T O ) . [ABSTRACT FROM AUTHOR]

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]

In this paper we determine the unique graph with minimal least eigenvalue (of the adjacency matrix) within the set of connected graphs of fixed order n and size m , whenever m = ⌈ n 2 ⌉ ⌊ n 2 ⌋ + a , where a is a fixed integral constant in [ 1 , ⌈ n 2 ⌉ − 1 ] . [ABSTRACT FROM AUTHOR]

Let k , p , q be positive integers with k < p < q + 1 . We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph K p , q of bipartition orders p and q by deleting k edges is attained when the deleted edges are all incident on a common vertex which is located in the partite set of order q . Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences. [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]

Abstract: A graph is called reflexive if its second largest eigenvalue does not exceed 2. In this paper, we determine all reflexive bipartite regular graphs. Any bipartite regular graph of degree at most 2 is reflexive as well as its bipartite complement. Apart from them, there is a finite number of resulting graphs. [Copyright &y& Elsevier]

Abstract: We derive some structural and spectral properties of regular bipartite graphs with three distinct non-negative eigenvalues. Next, we consider the relations between these graphs and two-class partially balanced incomplete block designs, and we present a number of situations when the graphs we consider are in fact the incidence graphs of those designs. As a consequence, we give a several constructions of connected regular bipartite graphs with six distinct eigenvalues, and we also determine all such graphs with degree 3, and all such graphs on at most 20 vertices. [Copyright &y& Elsevier]

Abstract: The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order. [Copyright &y& Elsevier]

Abstract: We determine all trees whose second largest eigenvalue does not exceed . Next, we consider two classes of bipartite graphs, regular and semiregular, with small number of distinct eigenvalues. For all graphs considered we determine those whose second largest eigenvalue is equal to . Some additional results are also given. [Copyright &y& Elsevier]

Abstract: The problem of identifying those simple, undirected graphs with vertices and edges that have the smallest minimum eigenvalue of the adjacency matrix is considered. Several general properties of the minimizing graphs are described. These strongly suggest bipartition, to the extent possible for the number of edges. In the bipartite case, the precise structure of the minimizing graphs is given for a number of infinite classes. [Copyright &y& Elsevier]

Abstract: In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices, and then obtain a lower bound for the least eigenvalue of a connected graph in terms of the number of cut vertices. During the discussion we also get some results for the spectral radius of a connected bipartite graph and its upper bound. [Copyright &y& Elsevier]

Abstract: In 1977, Ganter and Teirlinck proved that any matrix with nonzero elements can be partitioned into four submatrices of order of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any matrix with nonzero elements can be partitioned into submatrices of order of which at most contain nonzero elements. In 1995, Brualdi et al. showed that this conjecture is true if , or . They also found a counterexample of this conjecture for , , and . When , Rho showed that this conjecture is true if . When and , we show that this conjecture is true if or . As a result, we show that when , this conjecture is true if also. [Copyright &y& Elsevier]

Abstract: A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix which is in a prescribed way defined for any graph. This theory is called -theory. We outline a spectral theory of graphs based on the signless Laplacians and compare it with other spectral theories, in particular to those based on the adjacency matrix and the Laplacian . As demonstrated in the first part, the -theory can be constructed in part using various connections to other theories: equivalency with -theory and -theory for regular graphs, common features with -theory for bipartite graphs, general analogies with -theory and analogies with -theory via line graphs and subdivision graphs. In this part, we introduce notions of enriched and restricted spectral theories and present results on integral graphs, enumeration of spanning trees, characterizations by eigenvalues, cospectral graphs and graph angles. [Copyright &y& Elsevier]

Abstract: For , , a Bethe tree is a rooted tree with levels which the root vertex has degree , the vertices from level to have degree and the vertices at the level are pendent vertices. So et al, using a theorem by Ky Fan have obtained both upper and lower bounds for the Laplacian energy of bipartite graphs. We shall employ the above mentioned theorem to obtain new and improved bounds for the Laplacian energy in the case of Bethe trees. [Copyright &y& Elsevier]

Abstract: We say that two graphs and with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers such that the complete bipartite graph is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most linearly independent commuting adjacency matrices of size ; and if this bound occurs, then there exists a Hadamard matrix of order . Finally, we determine the centralizers of some families of graphs. [Copyright &y& Elsevier]

It is proved that any set of finite pairwise nonisomorphic weakly cospectral connected pseudographs, as well as any set of pseudodigraphs satisfying various additional conditions, is finite. [ABSTRACT FROM AUTHOR]