Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 10 4 2025 12 01 Leavitt path algebras for order prime Cayley graphs of finite groups 743 761 14697 10.22049/cco.2024.28966.1798 EN Sumanta Das Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata-700019, India 0000-0002-8155-2805 Mridul Kanti Sen Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata-700019, India Sunil Kumar Maity Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata-700019, India Journal Article 2023 09 01 In this paper, we generalize the concept of Cayley graphs associated to finite groups. The aim of this paper is the characterization of graph theoretic properties of new type of directed graph $\Gamma_P(G;S)$ and algebraic properties of Leavitt path algebra of order prime Cayley graph $O\Gamma(G;S)$, where $G$ is a finite group with a generating set $S$. We show that the Leavitt path algebra of order prime Cayley graph $L_K(O\Gamma(G;S))$ of a non trivial finite group $G$ with any generating set $S$ over a field $K$ is a purely infinite simple ring. Finally, we prove that the Grothendieck group of the Leavitt path algebra $L_K(\Gamma_P(D_n;S))$ is isomorphic to $\mathbb{Z}_{2n-1}$, where $D_n$ is the dihedral group of degree $n$ and $S=\left\{a, b\right\}$ is the generating set of $D_n$. https://comb-opt.azaruniv.ac.ir/article_14697_14ee3edcd14e7fd99b06359abd80d2a7.pdf