the reciprocal degree distance (rdd)‎, ‎defined for a connected graph $g$ as vertex-degree-weighted sum of the reciprocal distances‎, ‎that is‎, ‎$rdd(g) =sumlimits_{u,vin v(g)}frac{d_g(u)‎ + ‎d_g(v)}{d_g(u,v)}.$ the reciprocal degree distance is a weight version of the harary index‎, ‎just as the degree distance is a weight version of the wiener index‎. ‎in this paper‎, ‎we present exact formu...

let g be a simple connected graph and {v1, v2, …, vk} be the set of pendent (vertices ofdegree one) vertices of g. the reduced distance matrix of g is a square matrix whose (i,j)–entry is the topological distance between vi and vj of g. in this paper, we obtain the spectrumof the reduced distance matrix of thorn graph of g, a graph which obtained by attaching somenew vertices to pendent vertice...

In this paper we propose and study a new structural invariant for graphs, called distance-unbalancedness, as measure of how much graph is (un)balanced in terms distances. Explicit formulas are presented several classes well-known graphs. Distance-unbalancedness trees also studied. A few conjectures stated some open problems proposed.

Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

let $g$ be a connected graph, and let $d[g]$ denote the double graph of $g$. in this paper, we first derive closed-form formulas for different distance based topological indices for $d[g]$ in terms of that of $g$. finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

hansen et‎. ‎al.‎, ‎using the autographix software ‎package‎, ‎conjectured that the szeged index $sz(g)$ and the‎ ‎wiener index $w(g)$ of a connected bipartite graph $g$ with $n geq ‎4$ vertices and $m geq n$ edges‎, ‎obeys the relation‎ ‎$sz(g)-w(g) geq 4n-8$‎. ‎moreover‎, ‎this bound would be the best possible‎. ‎this paper offers a proof to this conjecture‎.