The Homomorphism Submodule Graph

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.

arXiv commentWithdrawn by the author after a careful re-examination revealed errors in the proofs/results. The paper is being withdrawn pending a thorough revision.