Abstract For a graph G , and two distinct vertices u v of let $$n_{G}(u,v)$$ n G ( u , v ) be the number that are closer in to than . Miklavi? Šparl (arXiv:2011.0...

motivated by the terminal wiener index‎, ‎we define the ashwini index $mathcal{a}$ of trees as‎ begin{eqnarray*}‎ % ‎nonumber to remove numbering (before each equation)‎ ‎mathcal{a}(t) &=& sumlimits_{1leq i‎&+& deg_{_{t}}(n(v_{j}))],‎ ‎end{eqnarray*}‎ ‎where $d_{t}(v_{i}‎, ‎v_{j})$ is the distance between the vertices $v_{i}‎, ‎v_{j} in v(t)$‎, ‎is equal to the length of the shortest path start...

Let G be a simple connected graph and {v_1,v_2,..., v_k} be the set of pendent (vertices of degree one) vertices of G. The reduced distance matrix of G is a square matrix whose (i,j)-entry is the topological distance between v_i and v_j of G. In this paper, we compute the spectrum of the reduced distance matrix of the generalized Bethe trees.

In this paper we prove that any distance-balanced graph $G$ with $Delta(G)geq |V(G)|-3$ is regular. Also we define notion of distance-balanced closure of a graph and we find distance-balanced closures of trees $T$ with $Delta(T)geq |V(T)|-3$.