Some familiar graphs on the rings of measurable functions
نویسندگان
1 Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, West Bengal, India.
2 Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, West Bengal, India.
3 Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, West Bengal, India.
doi
10.30504/jims.2025.403567.1128چکیده
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.
