On inverse symmetric division deg index of graphs

Abstract

One of the 148 discrete Adria indices is the symmetric division deg (SDD) index. It was developed about 13 years ago. Motivated by the success of the symmetric division deg index, Ghorbani et al. recently proposed an inverse version of this index, which they called the ISDD index (Inverse symmetric division deg index). The inverse symmetric division deg index (ISDD) of a graph Γ is defined as follows: $$ \mathrm{ISDD}(\mathrm{\Gamma })=\sum_{{v}_i{v}_j\in E(\mathrm{\Gamma })} \enspace \frac{{d}_i{d}_j}{{d}_i^2+{d}_j^2}, $$ where di is the degree of the vertex vi in Γ. In this paper, we determine the second maximal and the second minimal trees with respect to the inverse symmetric division deg index (ISDD). We prove that the star gives the minimal and the complete bipartite graph K⌈n/2⌉, ⌊n/2⌋ gives the maximal graphs with respect to the inverse symmetric division deg index (ISDD) among any chain graph of order n. Moreover, the Turán graph gives the maximal graph with respect to the ISDD index for any simple graph of order n with chromatic number k. Finally, we give concluding remarks about future works.