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arXiv · 2511.07837

The Homomorphism Submodule Graph

Shahram Mehry·Mansour Molaeinejad
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Abstract

Let $M$ be a left $R$-module. We define the \emph{homomorphism submodule graph} $\Gamma_{\mathrm{Hom}}(M)$ as the simple graph whose vertices are the proper submodules of $M$, with an edge between distinct vertices $N_1$ and $N_2$ if and only if $\mathrm{Hom}_R(N_1, M/N_2) \ne 0$ or $\mathrm{Hom}_R(N_2, M/N_1) \ne 0$. This graph encodes homological information about $M$ and reflects its internal structure. We compute $\Gamma_{\mathrm{Hom}}(M)$ for semisimple and uniserial modules, establish precise correspondences between graph-theoretic and algebraic properties, and prove that for modules over Artinian local rings, the isomorphism type of $M$ is determined by $\Gamma_{\mathrm{Hom}}(M)$. We also show that over commutative rings with identity, the graph is always chordal, and we relate its spectral radius to composition length in natural families.

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Shahram Mehry, Mansour Molaeinejad. 2025-11-11. The Homomorphism Submodule Graph. https://arxiv.org/abs/2511.07837

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