Milan Nath

On the distance spectral radius of trees

In this article, we prove that among trees on n vertices and matching number m, the dumbbell is 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.

On the spectra of graphs with edge-pockets

Let and be two vertex disjoint graphs of orders and respectively, where and Let be a specified edge of such that is isomorphic to Let be a subset of the edge set of and let denote the subgraph of induced by Let be the graph obtained by taking one copy of and vertex disjoint copies of and then pasting the edge in the th copy of with the edge where Then the copies of the graph that are pasted to the edges are called as edge-pockets, and we say is a graph with edge-pockets. In this article, we prove some results describing the Laplacian (resp. adjacency) spectrum of using the Laplacian (resp. adjacency) spectra of and The complete Laplacian (resp. adjacency) spectrum of is 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.

A NOTE ON THE DISTANCE SPECTRAL RADIUS OF SOME GRAPHS

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.

ON THE MINIMAL DISTANCE SPECTRAL RADIUS IN THE CLASS OF BICYCLIC GRAPHS

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 of graphs which contain induced ∞-graphs, and the class of graphs which contain induced θ-graphs. Bose et al. [2] have found the graph having the minimal distance spectral radius in . In this paper, we determine the graphs having the minimal distance spectral radius in . These results together give a complete characterization of the graphs having the minimal distance spectral radius in ℬn.

ON THE NULL-SPACES OF BICYCLIC SINGULAR GRAPHS

In [M. Nath and B. K. Sarma, On the null-spaces of unicyclic and acyclic graphs, Linear Algebra Appl.427 (2007) 42–54], Nath and Sarma gave an algorithm to find a basis for the null-space of a graph G when G is singular acyclic or unicyclic. In this paper, we find a basis for the null-space of G when G is a bicyclic singular graph.

Minimal configuration bicyclic graphs

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 C p and C q 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.

GRAPH TRANSFORMATION AND DISTANCE SPECTRAL RADIUS

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.

ON THE SECOND LARGEST SPECTRAL RADIUS OF UNICYCLIC BIPARTITE GRAPHS

In this paper, we determine the two graphs in the class of unicyclic bipartite graphs with n vertices, having the first two largest spectral radius.