On the graphs attaining the minimum for the multiplicative variable Euler--Sombor index

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Document Type : Original paper

Author

Department of Mathematics, Khansar Faculty, University of Isfahan Isfahan, Iran

Abstract

We introduce the multiplicative variable Euler–Sombor index, \[\Pi_{\mathrm{ES}_\lambda}(G) = \prod_{uv\in E(G)} \sqrt{d_u^2 + d_v^2 + \lambda d_u d_v}, \] where $\lambda$ is a real parameter. For $0\le \lambda < 2$, we characterize the graphs that attain the minimum of this quantity. A key intermediate result asserts that any connected graph attaining the minimum with fixed order and size must satisfy $\Delta(G)-\delta(G)\le 1$. This structural fact yields a complete description of the graphs that attain the minimum in several families: paths among trees, cycles among unicyclic graphs, and, for bicyclic graphs, the book and single-bridge dumbbell graphs. In the general setting of $k$-cyclic graphs (with $k\ge 3$ and $n\ge 5(k-1)$), the graphs that attain the minimum are precisely the bidegreed graphs with degree set $\{2,3\}$, and the value they attain is exactly
\[
(18+9\lambda)^{\frac{3k-4}{2}}(13+6\lambda)(8+4\lambda)^{\frac{n-2k+1}{2}}.
\]
Our framework provides the first extremal characterization for the multiplicative Euler–Sombor index (the case $\lambda=1$), together with a complete description for all $0\le \lambda<2$. This simultaneously extends the known results for the classical multiplicative Sombor index ($\lambda=0$) and introduces a new parametric family that has not been previously investigated.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 08 October 2026

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