University of Guilan Journal of Algebra and Related Topics 2345-3931 14 Special Issue- Dedicated to the memory of Jürgen Herzog (1941-2024). 2026 04 01 Join maximal element graph of lattice modules 115 123 9431 10.22124/jart.2026.28291.1704 EN L. Sherpa Department of Mathematics‎,‎ Savitribai Phule Pune University, Pune‎, ‎India V. Kharat Department of Mathematics‎,‎ Savitribai Phule Pune University, Pune‎, ‎India M. Agalave Department of Mathematics‎,‎ Fergusson College(Autonomus), Pune‎, ‎India N. M. Phadatare Bharati Vidyapeeth Deemed to be University College of Engineering, Pune‎, ‎India Journal Article 2024 08 27 Let $\pounds$ be a $C$-lattice and $M$ be a lattice module over $\pounds$‎. ‎The join maximal element graph $\mathbb{G}(M)$ is a simple‎, ‎undirected graph with all proper non-zero elements of $M$ as vertices‎, ‎and two distinct vertices‎, ‎$N$ and $K$‎, ‎are adjacent if and only if $N\vee K\in Max(M)$‎, ‎where $Max(M)$ is the collection of all maximal elements of $M$‎. ‎In this paper‎, ‎some properties of the graph $\mathbb{G}(M)$ like diameter‎, ‎girth and clique number are investigated‎. ‎Also‎, ‎the interplay between the algebraic properties of $M$ and the properties of those graphs is studied‎. ‎Maximal element Jacobson radical $J_{rad}(M)$ Join maximal element graph $\mathbb{G}(M)$ https://jart.guilan.ac.ir/article_9431_70964bf3a60bb5a4ff35d04635a81402.pdf