On the graphs of a fixed cyclomatic number and order with minimum general sum-connectivity and Platt indices

The general sum-connectivity and Platt indices of a graph G are defined by SCa(G)= n-ary sumation xy is an element of E(G)(dx+dy)a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {SC}}_a(G)=\sum _{xy\in E(G)}(d_x+d_y)<^>a$$\end{document} and Pla(G)= n-ary sumation xy is an element of E(G)(dx+dy-2)a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Pl}}_a(G)=\sum _{xy\in E(G)}(d_x+d_y-2)<^>a$$\end{document}, respectively, where a is a real number, E(G) indicates the edge set of G, and dv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_v$$\end{document} indicates the degree of a vertex v of G. The cyclomatic number of G is the least number of edges required to be deleted from G to make it acyclic. If the maximum degree of G is less than 5, then G is referred to as a molecular graph. In this paper, the problem of determining graphs possessing the minimum values of the indices SCa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {SC}}_a$$\end{document} and Pla\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Pl}}_a$$\end{document} among all connected (molecular) graphs of order n and cyclomatic number t is solved for 0<a<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<a<1$$\end{document} and n >= 2(t-1)>= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2(t-1)\ge 2$$\end{document} with n >= 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 4$$\end{document}. It is proved that the difference between the maximum and minimum degrees of the aforementioned extremal graphs is at most 1.