distance-balanced graphs are introduced as graphs in which every edge uv has the followingproperty: the number of vertices closer to u than to v is equal to the number of vertices closerto v than to u. basic properties of these graphs are obtained. in this paper, we study theconditions under which some graph operations produce a distance-balanced graph.

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 $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.

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...

the emph{harary index} $h(g)$ of a connected graph $g$ is defined as $h(g)=sum_{u,vin v(g)}frac{1}{d_g(u,v)}$ where $d_g(u,v)$ is the distance between vertices $u$ and $v$ of $g$. the steiner distance in a graph, introduced by chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. for a connected graph $g$ of order at least $2$ and $ssubseteq v(g)$, th...

Distance-balanced graphs are introduced as graphs in which every edge uv has the following property: the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Basic properties of these graphs are obtained. In this paper, we study the conditions under which some graph operations produce a distance-balanced graph.

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...

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 $Delta(T)geq |V(T)|-3$.