Eulerian and Clique number of the Zero Divisor Graph $\Gamma[ L(+)M]$

Abstract

The zero divisor graph of the idealization ring formed by combining rings L and M is represented as $\Gamma[L(+)M]$. A graph is created where the vertices represent the non-zero zero divisors of $L(+)M$,
two vertices are considered adjacent when the product of their values equals zero in this scenario. In this article, we investigate $\Gamma[{Z}_n(+)Z_{p_1}]$, where $n$ is equal, to the product of $p_1^rq_1$ for some positive integer. To find out when these graphs are Eulerian and more importantly, we are examining the clique number
of $\Gamma[{Z}_n(+)Z_{p_1}]$ for $n=p_1^rq_1$.