On the ring of real-valued measurable functions $\mathcal{M}(X,\mathcal{A})$‎, ‎we redefine the co-maximal graph $\Gamma'_2(\mathcal{M}(X,\mathcal{A}))$‎, ‎the annihilator graph $AG(\mathcal{M}(X,\mathcal{A}))$ and the weakly zero-divisor graph \linebreak $W\Gamma(\mathcal{M}(X,\mathcal{A}))$ with the help of a measure $\mu$ defined on the measurable space $(X,\mathcal{A})$‎. ‎First we observe that the vertex set of $\Gamma'_2(\mathcal{M}(X,\mathcal{A}))$ is equal to the vertex set of the zero-divisor graph $\Gamma(\mathcal{M}(X,\mathcal{A}))$ of $\mathcal{M}(X,\mathcal{A})$‎. ‎We show that‎, ‎$\Gamma'_2(\mathcal{M}(X,\mathcal{A}))$ and $\Gamma(\mathcal{M}(X,\mathcal{A}))$ are not isomorphic as graphs in general‎, ‎nevertheless we deduce a sufficient condition for them to be isomorphic as graphs‎. ‎We establish a condition for which $\Gamma'_2(\mathcal{M}(X,\mathcal{A}))$‎, ‎$\Gamma(\mathcal{M}(X,\mathcal{A}))$ and $AG(\mathcal{M}(X,\mathcal{A}))$ are equal‎. ‎We further characterise the non-atomicity of the measure space $(X,\mathcal{A},\mu)$ by some graph-theoretic properties of $\Gamma'_2(\mathcal{M}(X,\mathcal{A}))$ and $AG(\mathcal{M}(X,\mathcal{A}))$‎. ‎Moreover‎, ‎we realize that $W\Gamma(\mathcal{M}(X,\mathcal{A}))$ is a complete partite graph‎.