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Letandbe two vertex disjoint graphs of ordersandrespectively, whereandLetbe a specified edge ofsuch thatis isomorphic toLetbe a subset of the edge set ofand letdenote the subgraph ofinduced byLetbe the graph obtained by taking one copy ofandvertex disjoint copies ofand then pasting the edgein theth copy ofwith the edgewhereThen the copies of the graphthat are pasted to the edgesare called as edge-pockets, and we sayis a graph withedge-pockets. In this article, we prove some results describing the Laplacian (resp. adjacency) spectrum ofusing the Laplacian (resp. adjacency) spectra ofandThe complete Laplacian (resp. adjacency) spectrum ofis also described in some particular cases. As an application, we show that these results enable us to construct infinitely many pairs of Laplacian (resp. adjacency) cospectral graphs. [ABSTRACT FROM AUTHOR]

Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. The class of bicyclic graphs of order n, denoted by Bn, can be partitioned into two subclasses: the class B*n of graphs which contain induced co-graphs, and the class B**n of graphs which contain induced θ-graphs. Bose et al. [2] have found the graph having the minimal distance spectral radius in B*n. In this paper, we determine the graphs having the minimal distance spectral radius in B**n. These results together give a complete characterization of the graphs having the minimal distance spectral radius in Bn. [ABSTRACT FROM AUTHOR]

We characterize graphs with minimal distance spectral radius in two classes of graphs: with vertex connectivity k and minimum degree at least k, and with given number of blocks. Moreover, we determine the unique graph that maximizes the distance spectral radius among all graphs with given clique number. [ABSTRACT FROM AUTHOR]

Trees are very common in the theory and applications of combinatorics. In this paper, we consider graphs whose underlying structure is a tree and study the behavior of the distance spectral radius under a graph transformation. As an application, we find the corona tree that maximizes the distance spectral radius among all corona trees with a fixed maximum degree. We also find the graph with minimal (maximal) distance spectral radius among all corona trees. Finally, we determine the graph with minimal distance spectral radius in a special class of corona trees. [ABSTRACT FROM AUTHOR]

In this article, we prove that among trees onnvertices and matching numberm, the dumbbellis the unique tree that maximizes the distance spectral radius, which was conjectured by Aleksandar Ilić [Distance spectral radius of trees with given matching number, Discr. Appl. Math. 158 (2010), pp. 1799–1806]. In addition, we find the unique tree that maximizes the distance spectral radius in the class of trees with a given number of pendent vertices. [ABSTRACT FROM AUTHOR]

The nullity η(G) of a graph G is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. If η(G)= 1, then the core of G is the subgraph induced by the vertices associated with the nonzero entries of the kernel eigenvector. The set of vertices which are not in the core is the periphery of G. A graph G with nullity one is minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. An ∞-graph ∞(p, l, q) is a graph obtained by joining two vertex-disjoint cycles Cp and Cq by a path of length l≥ 0. Let B* be the class of bicyclic graphs with an ∞-graph as an induced subgraph. In this article, we characterize the graphs in B* which are minimal configurations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we determine the unique graph with minimum distance spectral radius among all connected bipartite graphs of order with a given matching number. Moreover, we characterize the graphs with minimal distance spectral radius in the class of all connected bipartite graphs with a given vertex connectivity. [Copyright &y& Elsevier]

Abstract: Let be the class of all connected graphs of order with pendent vertices. In this paper, we determine the unique graph with minimal distance spectral radius in In addition, we determine the unique graph with maximal distance spectral radius in for each [Copyright &y& Elsevier]

A graph G is singular with nullity η(G), if zero is an eigenvalue of its adjacency matrix with multiplicity η(G). If η(G) = 1, then the core of G is the subgraph induced by the vertices associated with the non-zero entries of the kernel eigenvector. The set of vertices which are not in the core is called the periphery of G. A graph G with nullity one is called a minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. In this article, we describe the structure of a singular unicyclic graph and single out the class of unicyclic graphs which are minimal configurations. [ABSTRACT FROM AUTHOR]