Iranian Journal of Mathematical Chemistry

Iranian Journal of Mathematical Chemistry

On the Inverse Sum Indeg Index and its Associated Energy

Document Type : Research Paper

Authors
Slobodan Filipovski 1
‎Mirza Red\v{z}i\'c 2
Elvis Barakovi\'c 3
1 Faculty of Mathematics‎, ‎Natural Sciences and Information Technologies‎, ‎University of Primorska‎, ‎Glagolja\v{s}ka 8‎, ‎SI-6000 Koper‎, ‎Slovenia
2 Karlsruhe Institute of Technology (KIT)‎, ‎76131 Karlsruhe‎, ‎Germany
3 Faculty of Natural Sciences and Mathematics‎, ‎University of Tuzla‎, ‎Univerzitetska 4‎, ‎75000 Tuzla‎, ‎Bosnia and Herzegovina
10.22052/ijmc.2026.257236.2033
Abstract
‎Let G be a simple graph with degree sequence $\{d_v\}$‎. ‎The inverse sum indeg (ISI) matrix S is defined by‎ $s_{uv} = \frac{d_u d_v}{d_u‎ + ‎d_v}$ if $uv \in E(G)$ and 0 otherwise‎. ‎The ISI energy of G is defined as‎ $\mathcal{E}_{ISI}(G) = \sum_{i=1}^n |\tau_i|$‎, ‎where $\tau_1,\dots,\tau_n$ are the eigenvalues of S. In this paper‎, ‎we establish new upper and lower bounds for the ISI index and ISI energy in terms of classical degree-based‎ graph invariants‎, ‎including the second Zagreb index‎, ‎geometric–arithmetic index‎, ‎harmonic index‎, ‎Sombor index‎, ‎and Randić index‎. Several of these results refine and improve the classical estimate $\mathcal{E}_{ISI}(G) \le \sqrt{2nB}$‎, ‎where‎ $B = \sum_{uv\in E(G)} \left(\frac{d_u d_v}{d_u‎ + ‎d_v}\right)^2$‎. We further derive spectral refinements involving extremal eigenvalues and the variance of the squared eigenvalues, and recall known determinant-based bounds. ‎In addition‎, ‎we obtain new bounds for the largest eigenvalue of S and establish further relations between ISI energy and‎ structural molecular descriptors‎.
Keywords
Inverse sum indeg index
ISI energy
Degree-based topological indices
Spectral graph theory
Graph energy
Subjects
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Iranian Journal of Mathematical Chemistry
Volume 17, Issue 4
In Progress
Autumn 2026
Pages 401-414