Table 1 Mathematical formulations of Sombor degree-based TIs.

Sombor index¹⁹

SO

\(\sqrt{d(u)^2+d(v)^2}\)

\(\mathcal {I}_{SO}\)

\(-\sum \limits _{uv\in E(\mathscr {G})}\frac{\sqrt{d(u)^2+d(v)^2}}{SO}\log {\frac{\sqrt{d(u)^2+d(v)^2}}{SO}}\)

Reduced Sombor index¹⁹

RSO

\(\sqrt{(d(u)-1)^2+(d(v)-1)^2}\)

\(\mathcal {I}_{RSO}\)

\(-\sum \limits _{uv\in E(\mathscr {G})}\frac{\sqrt{(d(u)-1)^2+(d(v)-1)^2}}{RSO}\log {\frac{\sqrt{(d(u)-1)^2+(d(v)-1)^2}}{RSO}}\)

Modified Sombor index³⁰

mSO

\(\frac{1}{\sqrt{d(u)^2+d(v)^2}}\)

\(\mathcal {I}_{mSO}\)

\(-\sum \limits _{uv\in E(\mathscr {G})}\frac{\frac{1}{\sqrt{d(u)^2+d(v)^2}}}{mSO}\log {\frac{\frac{1}{\sqrt{d(u)^2+d(v)^2}}}{mSO}}\)

Reduced Modified Sombor index³⁰

mRSO

\(\frac{1}{\sqrt{(d(u)-1)^2+(d(v)-1)^2}}\)

\(\mathcal {I}_{mRSO}\)

\(-\sum \limits _{uv\in E(\mathscr {G})}\frac{\frac{1}{\sqrt{(d(u)-1)^2+(d(v)-1)^2}}}{mRSO}\log {\frac{\frac{1}{\sqrt{(d(u)-1)^2+(d(v)-1)^2}}}{mRSO}}\)

First Banhatti-Sombor index³¹

\(BSO_1\)

\(\sqrt{\frac{1}{d(u)^2}+\frac{1}{d(v)^2}}\)

\(\mathcal {I}_{BSO_1}\)

\(-\sum \limits _{uv\in E(\mathscr {G})}\frac{\sqrt{\frac{1}{d(u)^2}+\frac{1}{d(v)^2}}}{BSO_1}\log {\frac{\sqrt{\frac{1}{d(u)^2}+\frac{1}{d(v)^2}}}{BSO_1}}\)

Second Banhatti-Sombor index³²

\(BSO_2\)

\(\dfrac{1}{\sqrt{\frac{1}{d(u)^2}+\frac{1}{d(v)^2}}}\)

\(\mathcal {I}_{BSO_2}\)

\(-\sum \limits _{uv\in E(\mathscr {G})}\frac{\dfrac{1}{\sqrt{\frac{1}{d(u)^2}+\frac{1}{d(v)^2}}}}{BSO_2}\log {\frac{\dfrac{1}{\sqrt{\frac{1}{d(u)^2}+\frac{1}{d(v)^2}}}}{BSO_2}}\)

Elliptic Sombor index³³

ESO

\((d(u)+d(v)){\sqrt{d(u)^2+d(v)^2}}\)

\(\mathcal {I}_{ESO}\)

\(-\sum \limits _{uv\in E(\mathscr {G})}\frac{(d(u)+d(v)){\sqrt{d(u)^2+d(v)^2}}}{ESO}\log {\frac{(d(u)+d(v)){\sqrt{d(u)^2+d(v)^2}}}{ESO}}\)

Euler Sombor index^34,35

EUSO

\(\sqrt{d(u)^2+d(v)^2+d(u)d(v)}\)

\(\mathcal {I}_{EUSO}\)

\(-\sum \limits _{uv\in E(\mathscr {G})}\frac{\sqrt{d(u)^2+d(v)^2+d(u)d(v)}}{EUSO}\log {\frac{\sqrt{d(u)^2+d(v)^2+d(u)d(v)}}{EUSO}}\)