Partial proof of Graham‎ ‎Higmanʹʹs conjecture related to coset diagrams

‎Higman has defined coset diagrams for‎ ‎$PSL(2,\mathbb{Z})$‎. ‎These diagrams are composed of fragments‎, ‎and the‎ ‎fragments are further composed of two or more circuits‎. ‎In $1983$‎, ‎Q‎. ‎Mushtaq has‎ ‎proved that existence of a certain fragment $\gamma $\ of a coset‎ ‎diagram in a coset diagram is a polynomial $f$\ in $\mathbb{Z}[z].$ Higman‎ ‎has conjectured that‎, ‎the polynomials related to the fragments are monic and‎ ‎for a fixed degree‎, ‎there are finite number of such polynomials‎. ‎In this‎ ‎paper‎, ‎we consider a family $\digamma $ of fragments such that each fragment‎ ‎in $\digamma $ contains one vertex $v$ fixed by‎ ‎\begin{equation*}‎ ‎F_{v}\left[ \left( xy^{-1}\right) ^{s_{1}}\left( xy\right) ^{s_{2}}\left(‎ ‎xy^{-1}\right) ^{s_{3}},\left( xy\right) ^{q_{1}}\left( xy^{-1}\right)‎ ‎^{q_{2}}\left( xy\right) ^{q_{3}}\right]‎ ‎\end{equation*}%‎ ‎where $s_{1},s_{2},s_{3},q_{1},q_{2},q_{3}\in‎ ‎%TCIMACRO{\U{2124}‎ }% ‎%BeginExpansion‎ ‎\mathbb{Z}‎ ‎%EndExpansion‎ ‎^{+},$ and prove this conjecture for the polynomials obtained from the‎ ‎fragments in $\digamma $‎.