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The Padmakar–Ivan index of a graph G is the sum over all edges uv of G of number of edges which are not equidistant from u and v. In this work, an exact expression for the PI index of the Cartesian product of bipartite graphs is computed. Using this formula, the PI indices of C 4 nanotubes and nanotori are computed.

The Szeged index is one of the most important topological indices defined in chemistry. In this paper, the Szeged index of the hexagonal triangle graph T(n) and the zigzag polyhex nanotube TUHC 6 [2p,q] are computed.

Let G be a connected graph, u; v be vertices of G and e = uv. The number of edges of G lying closer to u than to v is denoted by n eu (e|G) and the number of edges of G lying closer to v than to u is denoted by n ev (e|G). The PI polynomial of G is defined as PI(G; x) = N(u,v) {u,v} V(G) x , ⊆ ∑ where N(u,v) = n eu (e|G) + n ev (e|G), if e = uv; and = 0,… (More)

In this paper we prove that any distance-balanced graph G with ∆(G) ≥ |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 ∆(T) ≥ |V (T)| − 3.

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