  In this paper, replacing `equality' by 'equality almost everywhere' we modify
several terms associated with the ring of measurable functions defined on a
measure space $(X, \mathcal{A}, \mu)$ and thereby study the graph theoretic
features of the modified comaximal graph, annihilator graph and the weakly
zero-divisor graph of the said ring. The study reveals a structural analogy
between the modified versions of the comaximal and the zero-divisor graphs,
which prompted us to investigate whether these two graphs are isomorphic.
Introducing a quotient-like concept, we find certain subgraphs of the comaximal
graph and the zero-divisor graph of $\mathcal{M}(X, \mathcal{A})$ and show that
these two subgraphs are always isomorphic. Choosing $\mu$ as a counting
measure, we prove that even if these two induced graphs are isomorphic, the
parent graphs may not be so. However, in case of Lebesgue measure space on
$\mathbb{R}$, we establish that the comaximal and the zero-divisor graphs are
isomorphic. Observing that both of the comaximal and the zero-divisor graphs of
the ring $\mathcal{M}(X, \mathcal{A})$ are subgraphs of the annihilator graph
of the said ring, we find equivalent conditions for their equalities in terms
of the partitioning of $X$ into two atoms. Moreover, the non-atomicity of the
underlying measure space $X$ is characterized through graph theoretic phenomena
of the comaximal and the annihilator graph of $\mathcal{M}(X, \mathcal{A})$.
