On inverse symmetric division deg index of graphs

¹ Department of Mathematics, College of Sciences, University of Sharjah, 27272 Sharjah, UAE
² Department of Mathematics, Balurghat College, Balurghat 733101, India
³ Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

^* Corresponding author: kinkardas2003@gmail.com

Received: 8 September 2023
Accepted: 23 November 2023

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:

$$\mathrm{ISDD}\left(\mathrm{\Gamma }\right)=\sum _{v_i{v}_j\in E\left(\mathrm{\Gamma }\right)}\mathrm{} \frac{d_i{d}_j}{d_i^2+d_j^2},$$

where d_i is the degree of the vertex v_i 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.