the wiener polarity index wp(g) of a molecular graph g of order n is the number ofunordered pairs of vertices u, v of g such that the distance d(u,v) between u and v is 3. in anearlier paper, some extremal properties of this graph invariant in the class of catacondensedhexagonal systems and fullerene graphs were investigated. in this paper, some new bounds forthis graph invariant are presented....

In this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. Also, density and balanced irregular vague graphs are discussed and some of their properties are established. Finally we give an application of vague digraphs.

the wiener index $w(g)$ of a connected graph $g$‎ ‎is defined as $w(g)=sum_{u,vin v(g)}d_g(u,v)$‎ ‎where $d_g(u,v)$ is the distance between the vertices $u$ and $v$ of‎ ‎$g$‎. ‎for $ssubseteq v(g)$‎, ‎the {it steiner distance/} $d(s)$ of‎ ‎the vertices of $s$ is the minimum size of a connected subgraph of‎ ‎$g$ whose vertex set is $s$‎. ‎the {it $k$-th steiner wiener index/}‎ ‎$sw_k(g)$ of $g$ ...

For any graph G with fixed boundary there exists a layout in the plane, which minimizes the maximum Euclidean distance of any node to its neighbors. This layout balances the length of the graph edges and is therefore called a (length-) balanced layout of G. Furthermore the existence of a unique optimal balanced layout L with the following properties is proved: (i) L is the minimal element of an...

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 $...

Consider the problem of using a vehicle to transport k objects one at a time between s stations on a circular track. Let the cost of the transportation be the rotal distance traveled by the vehicle on the track. An 0 (k + M(s, q)) time algorithm is presented to find a minimum cost transportation, where M (m, n) is the time to solve a minimum spanning tree problem on a graph with m edges and n v...

the edge szeged index is a new molecular structure descriptor equal to the sum of products mu(e)mv(e) over all edges e = uv of the molecular graph g, where mu(e) is the number of edges which its distance to vertex u is smaller than the distance to vertex v, and nv(e) is defined analogously. in this paper, the edge szeged index of one-pentagonal carbon nanocone cnc5[n] is computed for the first ...

A perfect matching in a graph G is set of nonadjacent edges covering every vertex G. Motivated by recent progress on the relations between eigenvalues and number graph, this paper, we aim to present distance spectral radius condition guarantee existence matching. Let be an n-vertex connected where n even λ1(D(G)) Then following statements are true. (I) If 4≤n≤8 λ1(DG)≤λ1(D(Sn,n2−1)), then conta...

A balanced bipartite graph G is said to be 2p-Hamilton-biconnected if for any balanced subset W of size 2p of V (G), the subgraph induced by V (G)\W is Hamilton-biconnected. In this paper, we prove that “ Let p ≥ 0 and G be a balanced bipartite graph of order 2n with minimum degree δ(G) ≥ k, where n ≥ 2k − p + 2 and k ≥ p. If the number of edges e(G) > n(n − k + p− 1) + (k+2)(k−p+1), then G is ...