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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]

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]