The Reduced Smith Group of a Kneser Graph and Its Application to Group-Valued Magic Maps

For a finite abelian group $\mathrm{\Gamma }$, a map $f\text{\hspace{0.17em}⁣}:V(G)\to \mathrm{\Gamma }$ is a $\mathrm{\Gamma }$-magic map if the sum of its values over the neighbors of a vertex is independent of the vertex. It is affinely generating if its pairwise differences generate $\mathrm{\Gamma }$. When $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }=\mathrm{\mid }V(G)\mathrm{\mid }$, a bijective $\mathrm{\Gamma }$-magic map is a $\mathrm{\Gamma }$-distance magic labeling. For a regular graph $G$ with adjacency matrix $A$, write $\mathbf{1}$ for the all-ones vector indexed by $V(G)$, and let $\bar{A}$ denote the endomorphism induced by $A$ on ${\mathrm{\Lambda }}_{\mathbf{1}}={\mathbb{Z}}^{V(G)}\mathrm{/}\mathbb{Z}\mathbf{1}$. When $\bar{A}$ is nonsingular over $\mathbb{Q}$, define the reduced Smith group by ${\mathsf{S}}_{\mathrm{r}\mathrm{e}\mathrm{d}}(G)=\mathrm{coker}\bar{A}$. In previous work, we established that for every regular graph $G$ of positive degree with $\bar{A}$ nonsingular over $\mathbb{Q}$, one has $$G\text{ admits an affinely generating }\mathrm{\Gamma }\text{-magic map}⟺\mathrm{\Gamma }\hookrightarrow {\mathsf{S}}_{\mathrm{r}\mathrm{e}\mathrm{d}}(G)\mathrm{.}$$ We determine this group explicitly for Kneser graphs. If $r\ge 1$, $n\ge 2r$, and $m_j=\left(\genfrac{}{}{0px}{}{n}{j}\right)-\left(\genfrac{}{}{0px}{}{n}{j-1}\right)$, then $${\mathsf{S}}_{\mathrm{r}\mathrm{e}\mathrm{d}}(K(n,r))\cong \underset{j=1}{\overset{r}{⨁}}{\left(\mathbb{Z}\mathrm{/}\left(\genfrac{}{}{0px}{}{n-r-j}{r-j}\right)\mathbb{Z}\right)}^{m_j}\mathrm{.}$$ When the labeling group and the Kneser graph have the same order, the resulting embedding criterion leaves only one case in which an affinely generating map exists. More precisely, if $\mathrm{\Gamma }$ is an abelian group of order $\left(\genfrac{}{}{0px}{}{n}{r}\right)$, then $K(n,r)$ admits an affinely generating $\mathrm{\Gamma }$-magic map if and only if $(n,r)=(9,2)$ and $\mathrm{\Gamma }\cong \mathbb{Z}\mathrm{/}6\mathbb{Z}\oplus \mathbb{Z}\mathrm{/}6\mathbb{Z}$. A weak-Sidon-set bound shows that the affinely generating maps in the exceptional case cannot be bijective. Hence $K(n,r)$ admits no $\mathrm{\Gamma }$-distance magic labeling throughout the range $r\ge 1$ and $n\ge 2r$.