Some new relations have been established between Wiener indices, stability numbers and clique numbers for several classes of composite graphs that arise via graph products. For three of considered operations we show that they make a multiplicative pair with the clique number. 2010 Mathematics Subject Classification: 05C40, 05C90

The Wiener index is one of the main descriptors that correlate a chemical compound’s molecular graph with experimentally gathered data regarding the compound’s characteristics. A long standing conjecture on the Wiener index ([4], [5]) states that for any positive integer n (except numbers from a given 49 element set), one can find a tree with Wiener index n. In this paper, we prove that every i...

The distance d(u, v|G) between the vertices u and v of a molecular graph G is the length of a shortest u, v-path. We consider a class of Wiener–type topological indices Wλ(G) , defined as the sum of the terms d(u, v|G) λ over all pairs of vertices of G . Several special cases of Wλ(G) , namely for λ = +1 (the original Wiener number) as well as for λ = −2,−1,+1/2,+2 and +3 , were previously stud...

Let $G$ be a finite group. The intersection graph of is whose vertex set the all proper non-trivial subgroups and two distinct vertices $H$ $K$ are adjacent if only $H\cap K \neq \{e\}$, where $e$ identity group $G$. In this paper, we investigate some properties exploring topological indices such as Wiener, Hyper-Wiener, first second Zagreb, Schultz, Gutman eccentric connectivity $D_{2n}$ for $...

Let G be a connected graph and ξ(G) = Sze(G)−We(G), where We(G) denotes the edge Wiener index and Sze(G) denotes the edge Szeged index of G. In an earlier paper, it is proved that if T is a tree then Sze(T ) = We(T ). In this paper, we continue our work to prove that for every connected graph G, Sze(G) ≥ We(G) with equality if and only if G is a tree. We also classify all graphs with ξ(G) ≤ 5. ...

Given a tree T = (V,E), the second Zagreb index of T is denoted by M2(T ) = ∑ uv∈E d(u)d(v) and the Wiener polarity index of T is equal to WP (T ) = ∑ uv∈E(d(u)−1)(d(v)−1). Let π = (d1, d2, ..., dn) and π′ = (d1, d2, ..., dn) be two different non-increasing tree degree sequences. We write π π′, if and only if ∑n i=1 di = ∑n i=1 d ′ i, and ∑j i=1 di ≤ ∑j i=1 d ′ i for all j = 1, 2, ..., n. Let Γ...

The hyper-Kirchhoff index is introduced when the hyper-Wiener operator is applied to the resistance-distance matrix of a connected graph. We give lower and upper bounds for the hyper-Kirchhoff index, and determine the n-vertex unicyclic graphs with the smallest, the second and the third smallest as well as the largest, the second and the third largest hyper-Kirchhoff indices for n ≥ 5. We also ...