A vertex v ∈ V (G) is said to distinguish two vertices x, y ∈ V (G) of a graph G if the distance from v to x is di erent from the distance from v to y. A set W ⊆ V (G) is a total resolving set for a graph G if for every pair of vertices x, y ∈ V (G), there exists some vertex w ∈ W − {x, y} which distinguishes x and y, while W is a weak total resolving set if for every x ∈ V (G)−W and y ∈ W, there exists some w ∈ W −{y} which distinguishes x and y. A weak total resolving set of minimum cardinality...

The Wiener index of a connected graph G, denoted by W(G), is defined as $½{\sum }_{u,v\in V\left(G\right)}{d}_G(u,v)$. Similarly, the hyper-Wiener index of a connected graph G, denoted by WW(G), is defined as $½W\left(G\right)+¼{\sum }_{u,v\in V\left(G\right)}d{²}_G(u,v)$. The vertex Padmakar-Ivan (vertex PI) index of a graph G is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. In this paper, the exact formulae for Wiener, hyper-Wiener and vertex PI indices of the strong product $G⊠K_{m₀,m₁,...,m_{r-1}}$, where $K_{m₀,m₁,...,m_{r-1}}$ is the complete multipartite graph with partite sets of sizes...

The Wiener index, W, is the sum of distances between all pairs of vertices in a graph G. The quadratic line graph is defined as L(L(G)), where L(G) is the line graph of G. A generalized star S is a tree consisting of Δ ≥ 3 paths with the unique common endvertex. A relation between the Wiener index of S and of its quadratic graph is presented. It is shown that generalized stars having the property W(S) = W(L(L(S)) exist only for 4 ≤ Δ ≤ 6. Infinite families of generalized stars with this property...

The Wiener index of a connected graph is defined as the sum of the distances between all unordered pairs of its vertices. We characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut-vertices.

The Wiener index, denoted by W(G), of a connected graph G is the sum of all pairwise distances of vertices of the graph, that is, $W\left(G\right)=½{\Sigma }_{u,v\in V\left(G\right)}d(u,v)$. In this paper, we obtain the Wiener index of the tensor product of a path and a cycle.