For a (molecular) graph, the first Zagreb index is equal to the sum of squares of the degrees of vertices, and the F-index is equal to the sum of cubes of the degrees of vertices. In this paper, we introduce sixty four new operations on graphs and study the first Zagreb index and F-index of the resulting graphs.
Let Eβ (G) be the set of paths of length β in a graph G. For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index is defined as$$\begin{array}{}
\displaystyle
^\beta\chi_\alpha(G)=\sum \limits_{v_1v_2 \cdot \cdot \cdot v_{\beta+1}\in E_\beta(G)}(d_{G}(v_1)d_{G}(v_2)...d_{G}(v_{\beta+1}))^{\alpha}.
\end{array}$$The (2,1)-connectivity index shows good correlation with acentric factor of an octane isomers. In this paper, we compute the (2, α)-connectivity index of certain class of graphs, present the upper and lower bounds for (2, α)-connectivity index in terms of number of vertices, number of edges and minimum vertex degree and determine the extremal graphs which achieve the bounds. Further, we compute the (2, α)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p,q], tadpole graphs, wheel graphs and ladder graphs.
In this paper, we obtain the expressions for forgotten topological index, hyper-Zagreb index and coindex for generalized transformation graphs Gaband their complements \overline/Gab.
The total line-cut graph of a graph G = (V,E), denoted by TLc(G), is the graph with point set E(G) ∪ W(G), where W(G) is the set of cutpoints of G, in which two points are adjacent if and only if they correspond to adjacent lines of G or correspond to adjacent or coadjacent cutpoints of G or one point corresponds to a line e of G and the other corresponds to a cutpoint c of G such that e is incident with c. In this paper, we offer a structural characterization of total line-cut graphs.
We consider the generalized transformation graphs G ab and obtain expressions for their first and second Zagreb indices and coindices. Analogous expressions are obtained also for the complements of G ab .
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