Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Articles in Press 2025 10 17 Join standard graph of a lattice 15031 10.22049/cco.2025.30680.2577 EN N Aishwarya Nayak Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India Pallavi P Department of Mathematics, Manipal Institute of Technology Bengaluru, Manipal Academy of Higher Education, Manipal, India Syam Prasad Kuncham Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India Tapatee S Department of Mathematics, Manipal Institute of Technology Bengaluru, Manipal Academy of Higher Education, Manipal, India Harikrishnan P K Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India Journal Article 2025 06 01 In this paper, we introduce and investigate the join standard graph $G_S(L)$ of a finite lattice $L$. We explore structural properties of the graph such as connectedness, girth, and provide necessary and sufficient conditions for the existence of universal and isolated vertices. We show that a lattice homomorphism $\varphi$ from $L_1$ to $L_2$ induces a graph homomorphism between $G_S(L_1)$ and $G_S(L_2)$. We further analyze the relationship between the graph of a lattice product and the product of graphs of its constituent lattices. Subsequently, we establish a condition under which the graph becomes hypertriangulated. We prove that the graph $G_S(L)$ is complemented if and only if the underlying lattice has cardinality at most two. Finally, we provide a criterion under which the subgraph $G_S(L)-1$ becomes disconnected. https://comb-opt.azaruniv.ac.ir/article_15031_c1638e5c815b3177fa7afcf5a455d80d.pdf