The Eigensharp Property for Unit Graphs Associated with Some Finite Rings

Abstract

Let R be a commutative ring with unity. The unit graph G(R) is defined such that the vertex set of G(R) is the set of all elements of R, and two distinct vertices are adjacent if their sum is a unit in R. In this paper, we show that for each prime, p,G(Zp) and G(Z2p) are eigensharp graphs. Likewise, we show that the unit graph associated with the ring Zp[x]∕x2 is an eigensharp graph.

Keywords

• commutative ring
• unit graph
• graph join
• biclique
• biclique partition number
• eigensharp graph