Semisymmetric Graphs Defined by Finite-Dimensional Generalized Kac–Moody Algebras

Let Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q$$\end{document} be a finite field with q not powers of 2 or 3. In this paper, we mainly clarify whether the Lie graph defined by a finite-dimensional generalized Kac–Moody algebra is Cayley graph or not. More precisely, we prove that the graph L(M1,3,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}(M_1,3,q)$$\end{document} is a Cayley graph on a group of dihedral type, neither L(M2,4,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}(M_2,4,q)$$\end{document} nor L(M3,6,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}(M_3,6,q)$$\end{document} is vertex transitive. Moreover, L(M1,3,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}(M_1,3,q)$$\end{document} is a Hamiltonian graph. As an application, we construct an infinite family of semisymmetric (i.e., regular, edge-transitive and non-vertex-transitive graphs) graphs.