Subspace Sum Graphs over Residue Class Rings

Let \(\mathbb{Z}_{p^{s}}\) denote the residue class ring modulo \(p^{s}\) ,where \(p\) is a prime number and \(s\) is a positive integer. In thispaper, we introduce the subspace sum graph over \(\mathbb{Z}_{p^{s}}\) , which is a graph whose vertices arenon-trivial proper subspaces of \(\mathbb{Z}_{p^{s}}^{n}\) (for \(n\geq 2\) ). Two distinct vertices \(X\) and \(Y\) are adjacent if and only if \(\textrm{rank}\left(\begin{matrix}G_{X}\\ G_{Y}\end{matrix}\right)=n\) , where \(G_{X}\) and \(G_{Y}\) are generatormatrices for \(X\) and \(Y\) , respectively. We determine variousproperties of \(\mathcal{G}(\mathbb{Z}_{p^{s}}^{n})\) , including itsorder, vertex degrees, diameter, girth, clique number, andchromatic number.