Narrow Results

As we know that some chemically interesting graphs can be obtained by different graph operations on some general or particular graphs, it is important to study such graph operations in order to understand how it is related to the corresponding topological indices of the original graphs. In [A. Miličević, A. Nikolić and S. Trinajstić, On reformulated Zagreb indices, Mol. Divers.8 (2004) 393–399], authors defined two new variants of Corona product and investigated some topological indices. In this paper, we extended the work and found the formulas of the F-index, hyper-Zagreb index, and Sigma index for double join of graphs based on the total graph and double corona of graphs based on the total graph. Moreover, some applications of the derived results are also discussed.

Let $G$ be a graph with $n$ vertices, and let $d_u$ be the degree of the vertex $u$ in the graph $G$. The Randić matrix of $G$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to ${(d_u{d}_v)}^{\frac{-1}{2}}$ if the vertex $u$ and the vertex $v$ of $G$ are adjacent, and $0$ otherwise. The Randić eigenvalues of $G$ are the eigenvalues of its Randić matrix and the Randić energy of $G$ is the sum of the absolute values of its Randić eigenvalues. In this paper, we obtain some new results for the Randić eigenvalues and the Randić energy of a graph.

For a molecular graph $G$, the $F$-index or forgotten topological index is defined as the sum of cubes of degree of all vertices of the graph and the hyper-Zagreb index is equal to the sum of square of sum of degree of all adjacent vertices of the graph. In this paper, we obtain $F$-index, hyper-Zagreb index and their coindices of ${ℱ}$-tensor products (four new tensor products based on transformations of a graph) of graphs and their complements.