Somnath Paul

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 MAXIMAL DISTANCE SPECTRAL RADIUS IN A CLASS OF BICYCLIC GRAPHS

Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. Let Pp+1 = x1x2⋯xp+1, Pt+1 = y1y2⋯yt+1 and Pq+1 = z1z2⋯zq+1 be three vertex-disjoint paths. Identifying the initial vertices as u0 and the terminal vertices as v0, the resultant graph, denoted by θ(p; t; q), is called a θ-graph. Let

On the spectra of graphs with edge-pockets

\mathcal{B}_{n}

A NOTE ON THE DISTANCE SPECTRAL RADIUS OF SOME GRAPHS

be the class of all bicyclic graphs on n vertices, which contain a θ-graph as an induced subgraph. In this paper, we study the distance spectral radius of bicyclic graphs in

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

\mathcal{B}_{n}

On distance and distance Laplacian spectra of corona of two graphs

, and determine the graph with the largest distance spectral radius.

GRAPH TRANSFORMATION AND DISTANCE SPECTRAL RADIUS

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.

Distance spectral radius of graphs with r pendent vertices

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 distance spectral radius of bipartite 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 distance Laplacian spectra of graphs

Corona of two graphs has been defined in [F. Harary, Graph Theory (Addison-Wesley, 1969)]. In this paper, we study the distance and the distance Laplacian spectra of corona of two graphs and describe the complete distance (distance Laplacian) spectrum for some particular cases. As an application, we show that the corona operation can be used to create distance singular graphs. We also show that these results enable us to construct infinitely many pairs of distance (respectively, distance Laplacian) cospectral graphs. Last, we give a graph transformation and discuss its effect on the distance Laplacian spectral radius.