An exponential dominating set of graph $G = (V,E )$ is a subset $Ssubseteq V(G)$ such that $sum_{uin S}(1/2)^{overline{d}{(u,v)-1}}geq 1$ for every vertex $v$ in $V(G)-S$, where $overline{d}(u,v)$ is the distance between vertices $u in S$ and $v  in V(G)-S$ in the graph $G -(S-{u})$. The exponential domination number, $gamma_{e}(G)$, is the smallest cardinality of an exponential dominating set....

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...

‎The Steiner distance of a graph‎, ‎introduced by Chartrand‎, ‎Oellermann‎, ‎Tian and Zou in 1989‎, ‎is a natural generalization of the‎ ‎concept of classical graph distance‎. ‎For a connected graph $G$ of‎ ‎order at least $2$ and $Ssubseteq V(G)$‎, ‎the Steiner‎ ‎distance $d(S)$ among the vertices of $S$ is the minimum size among‎ ‎all connected subgraphs whose vertex sets contain $S$‎. ‎Let $...

a set $wsubseteq v(g)$ is called a resolving set for $g$, if for each two distinct vertices $u,vin v(g)$ there exists $win w$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. the minimum cardinality of a resolving set for $g$ is called the metric dimension of $g$, and denoted by $dim(g)$. in this paper, it is proved that in a connected graph $...

The distance $d(u,v)$ between two vertices $u$ and $v$ of a graph $G$ is equal to the length of a shortest path that connects $u$ and $v$. Define $WW(G,x) = 1/2sum_{{ a,b } subseteq V(G)}x^{d(a,b) + d^2(a,b)}$, where $d(G)$ is the greatest distance between any two vertices. In this paper the hyper-Wiener polynomials of the Cartesian product, composition, join and disjunction of graphs are compu...