Table 1.

Description and derivation of various TI’s obtained from the M-polynomial and NM-polynomial where $D_x=x(\frac{\partial (z(x,y))}{\partial x})$, $D_y=y(\frac{\partial (z(x,y))}{\partial y})$, $S_x={\int }_0^x\frac{z(t,y)}{t}dt$, $S_y={\int }_0^y\frac{z(x,t)}{t}dt$, $J(z(x,y))=z(x,x)$, $Q_{\alpha }(z(x,y))=x^{\alpha }z(x,y)$ For D based TI’s: $\mathrm{\Delta }(u)=d_u$, $\mathrm{\Delta }(v)=d_v$, $z(x,y)=M(\mathcal{G};x,y)$ for NBD based TI’s: $\mathrm{\Delta }(u)=nd_u^{\ast }$, $\mathrm{\Delta }(v)=nd_v^{\ast }$, $z(x,y)=NM_{}^{\ast }(\mathcal{G};x,y)$.

Topological index (TI)	Formula $g(\mathrm{\Delta }(u),\mathrm{\Delta }(v))$	Derivation from z(x,y) = $M(\mathcal{G};x,y)$ or $NM_{}^{\ast }(\mathcal{G};x,y)$	References of TI’s first introduced:
$1^{\mathit{st}}$ Zagreb index: $M_1(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}(\mathrm{\Delta }(u)+\mathrm{\Delta }(v))$	$(D_x+D_y){(z(x,y))}_{x=y=1}$	44
$3^{\mathit{rd}}$ version Zagreb index: $NM_1^{\ast }(\mathcal{G})$
$2^{\mathit{nd}}$ Zagreb index: $M_2(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}(\mathrm{\Delta }(u)\times \mathrm{\Delta }(v))$	$(D_x\times D_y){(z(x,y))}_{x=y=1}$	44
Nbd $2^{\mathit{nd}}$ zagreb index: $NM_2^{\ast }(\mathcal{G})$
$2^{\mathit{nd}}$ modified Zagreb index: $m{M}_2(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}\frac{1}{\mathrm{\Delta }(u)\mathrm{\Delta }(v)}$	$(S_x{S}_y){(z(x,y))}_{x=y=1}$	45
Nbd $2^{\mathit{nd}}$ modified zagreb index: $NmM_2^{\ast }(\mathcal{G})$
Redefined $3^{\mathit{rd}}$ Zagreb index: $ReZ{G}_3(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}[\mathrm{\Delta }(u)\mathrm{\Delta }(v)\times (\mathrm{\Delta }(u)+\mathrm{\Delta }(v))]$	$D_x{D}_y(D_x+D_y){(z(x,y))}_{x=y=1}$	46,47
$3^{\mathit{rd}}$ Nde index: $ND_3^{\ast }(\mathcal{G})$
Forgotten topological index: $F(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}({\mathrm{\Delta }}^2(u)+{\mathrm{\Delta }}^2(v))$	$(D_x^2+D_y^2){(z(x,y))}_{x=y=1}$	48
Nbd Forgotten topological index: $NF_{}^{\ast }(\mathcal{G})$
Randic index: $R_{\alpha }(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}{(\mathrm{\Delta }(u)\mathrm{\Delta }(v))}^{\alpha }$	$(D_x^{\alpha }{D}_y^{\alpha }){(z(x,y))}_{x=y=1}$	49
Nbd Randic index: $NR_{\alpha }^{\ast }(\mathcal{G})$
Inverse Randic index: $R{R}_{\alpha }(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}\frac{1}{{(\mathrm{\Delta }(u)\mathrm{\Delta }(v))}^{\alpha }}$	$(S_x^{\alpha }{S}_y^{\alpha }){(z(x,y))}_{x=y=1}$	50
Nbd inverse Randic index: $NRR_{\alpha }^{\ast }(\mathcal{G})$
Symmetric Division index: $SDD(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}\frac{{\mathrm{\Delta }}^2(u)+{\mathrm{\Delta }}^2(v)}{\mathrm{\Delta }(u)\mathrm{\Delta }(v)}$	$(D_x{S}_y+S_x{D}_y){(z(x,y))}_{x=y=1}$	51
Fifth NDe index: $ND_5^{\ast }(\mathcal{G})$
Harmonic index: $H(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}\frac{2}{\mathrm{\Delta }(u)+\mathrm{\Delta }(v)}$	$(2S_{x}J){(z(x,y))}_{x=1}$	52
Nbd Harmonic index: $NH_{}^{\ast }(\mathcal{G})$
Inverse sum indeg index: $I(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}\frac{\mathrm{\Delta }(u)\mathrm{\Delta }(v)}{\mathrm{\Delta }(u)+\mathrm{\Delta }(v)}$	$(S_{x}J{D}_x{D}_y){(z(x,y))}_{x=1}$	51
Nbd Inverse sum indeg index: $NI_{}^{\ast }(\mathcal{G})$
Augmented Zagreb index: $A(\mathcal{G})$	${\sum }_{uv\in E(\mathcal{G})}^{\phantom{i}}\{\frac{\mathrm{\Delta }(u)\mathrm{\Delta }(v)}{\mathrm{\Delta }(u)+\mathrm{\Delta }(v)-2}{\}}^3$	$(S_x^3{Q}_{-2}J{D}_x^3{D}_y^3){(z(x,y))}_{x=1}$	53,54
Sanskurti index: $S_{}^{\ast }(\mathcal{G})$