lower and upper bounds between energy‎, ‎laplacian energy‎, ‎and sombor index of some graphs

‎ivan gutman has introduced two essential indices; the energy of a graph g‎, ‎and the sombor index of that‎. ‎$varepsilon(g)$‎, ‎which stands for the first index‎, ‎is the sum of the absolute values of all eigenvalues related to the adjacency matrix of the graph $g$‎. ‎the second‎, ‎defined as $so(g)=sum _{uv in e(g)}sqrt{d_u^2+d_v^2}$‎, ‎where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v$ in $g$‎, ‎respectively‎. ‎it was proved that if $g$ is a graph of order at least 3‎, ‎then $varepsilon(g)leq so(g)$ and if $g$ is a connected graph of order $n$ that is not $p_n$ for $nleq 8$‎, ‎then $varepsilon(g)leq frac{so(g)}{2}$‎.‎in this paper‎, ‎we have strengthened these results and will obtain several lower and upper bounds between the energy of a graph‎, ‎laplacian energy‎, ‎and the sombor index‎.