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The generalized power of a simple graph G , denoted by G k , s , is obtained from G by blowing up each vertex into an s -set and each edge into a k -set, where 1 ≤ s ≤ k 2 . When s < k 2 , G k , s is always odd-bipartite. It is known that G k , k 2 is non-odd-bipartite if and only if G is non-bipartite, and G k , k 2 has the same adjacency (respectively, signless Laplacian) spectral radius as G . In this paper, we prove that, regardless of multiplicities, the H -spectrum of A ( G k , k 2 ) (respectively, Q ( G k , k 2 ) ) consists of all eigenvalues of the adjacency matrices (respectively, the signless Laplacian matrices) of the connected induced subgraphs (respectively, modified induced subgraphs) of G . As a corollary, G k , k 2 has the same least adjacency (respectively, least signless Laplacian) H -eigenvalue as G . We also discuss the limit points of the least adjacency H -eigenvalues of hypergraphs, and construct a sequence of non-odd-bipartite hypergraphs whose least adjacency H -eigenvalues converge to − 2 + 5 . [ABSTRACT FROM AUTHOR]

Let $$\fancyscript{B}(n,r)$$ be the set of all bicyclic graphs with $$n$$ vertices and $$r$$ cut edges. In this paper we determine the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in $$\fancyscript{B}(n,r)$$ . [ABSTRACT FROM AUTHOR]

Abstract: Let G be a graph and let be the adjacency matrix of G. The signature of G is the difference between the positive inertia index and the negative inertia index of . Ma et al. (2013) [10] conjectured that , where and respectively denote the number of cycles in G which have length and for some integers , and proved the conjecture holds for trees, unicyclic or bicyclic graphs. It is known that if G is bipartite, and the signature is closely related to the odd cycles or nonbipartiteness of a graph from the existing results. In this paper we show that the conjecture holds for line graphs and power trees. [Copyright &y& Elsevier]

Letbe a signed graph and letbe the adjacency matrix of. The nullity ofis the multiplicity of eigenvalue zero in the spectrum of. In this paper, we characterize the signed graphs of orderwith nullityor, and introduce a graph transformation which preserves the nullity. As an application we determine the unbalanced bicyclic signed graphs of orderwith nullityor, and signed bicyclic graphs (including simple bicyclic graphs) of orderwith nullity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce the nullity of signed graphs, and give some results on the nullity of signed graphs with pendant trees. We characterize the unicyclic signed graphs of order n with nullity respectively. [Copyright &y& Elsevier]

Abstract: A connected graph with a unique cycle is called a unicyclic graph. A unicyclic graph with depth one may be thought of as being obtained from a cycle by appending pendent edges on each vertex in the cycle (for some integer ), denoted by . In this paper, we give a complete characterization on the nullity of the line graph as follows: Let . Then [(i)] if and only if and . [(ii)] if and only if and either [(1)] for , the length of any zero chain of is even and ; or [(2)] and one of and must hold. [(iii)] , otherwise. [Copyright &y& Elsevier]

Abstract: We investigate how the least eigenvalue of the signless Laplacian of a graph changes by relocating a bipartite branch from one vertex to another vertex, and minimize the least eigenvalue of the signless Laplacian among the class of connected graphs with fixed order which contains a given non-bipartite graph as an induced subgraph. [Copyright &y& Elsevier]

Abstract: In this paper, we characterize the graphs with maximum signless Laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity. We also discuss the minimum signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity. Consequently we give an upper bound of signless Laplacian or adjacency spectral radius of graphs in terms of connectivity. In addition we confirm a conjecture of Aouchiche and Hansen involving adjacency spectral radius and connectivity. [Copyright &y& Elsevier]

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: 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: Let be a mixed graph and let be the Laplacian matrix of the graph . The first eigenvalue and the first eigenvectors of are respectively referred to the least nonzero eigenvalue and the corresponding eigenvectors of . In this paper we focus on the properties of the first eigenvalue and the first eigenvectors of a nonsingular unicyclic mixed graph (abbreviated to a NUM graph). We introduce the notion of characteristic set associated with the first eigenvectors, and then obtain some results on the sign structure of the first eigenvectors. By these results we determine the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order and fixed girth, and the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order. [Copyright &y& Elsevier]

Let G be a complex unit gain graph which is obtained from an undirected graph Γ by assigning a complex unit φ ( v i v j ) to each oriented edge v i v j such that φ ( v i v j ) φ ( v j v i ) = 1 for all edges. The Laplacian matrix of G is defined as L ( G ) = D ( G ) − A ( G ) , where D ( G ) is the degree diagonal matrix of Γ and A ( G ) = ( a i j ) has a i j = φ ( v i v j ) if v i is adjacent to v j and a i j = 0 otherwise. In this paper, we provide a combinatorial description of det ( L ( G ) ) that generalizes that for the determinant of the Laplacian matrix of a signed graph. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce a new parameter for a graph called the minimum universal rank. This parameter is similar to the minimum rank of a graph. For a graph G the minimum universal rank of G is the minimum rank over all matrices of the form where A is the adjacency matrix of G, J is the all ones matrix and D is the matrix with the degrees of the vertices in the main diagonal, and are scalars. Bounds for general graphs based on known graph parameters are given, as is a formula for the minimum universal rank for regular graphs based on the multiplicity of the eigenvalues of A. The exact value of the minimum universal rank of some families of graphs are determined, including complete graphs, complete bipartite graph, paths and cycles. Bounds on the minimum universal rank of a graph obtained by deleting a single vertex are established. It is shown that the minimum universal rank is not monotone on induced subgraphs, but bounds based on certain induced subgraphs, including bounds on the union of two graphs, are given. [Copyright &y& Elsevier]