Join standard graph of a lattice

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India

2 Department of Mathematics, Manipal Institute of Technology Bengaluru, Manipal Academy of Higher Education, Manipal, India

Abstract

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.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 17 October 2025

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