Properties of graphs of neural codes

Abstract

A neural code on $ n $ neurons is a collection of subsets of the set $ [n]=\{1,2,\dots,n\} $. In this paper, we study some properties of graphs of neural codes. In particular, we study codeword containment graph (CCG) given by Chan et al. (SIAM J. on Dis. Math., 37(1):114-145,2017) and general relationship graph (GRG) given by Gross et al. (Adv. in App. Math., 95:65-95, 2018). We provide a sufficient condition for CCG to be connected. We also show that the connectedness and completeness of CCG are preserved under surjective morphisms between neural codes defined by A. Jeffs (SIAM J. on App. Alg. and Geo., 4(1):99-122,2020). Further, we show that if CCG of any neural code $\mathcal{C}$ is complete with $|\mathcal{C}|=m$, then $\mathcal{C} \cong \{\emptyset,1,12,\dots,123\cdots m\}$ as neural codes. We also prove that a code whose CCG is complete is open convex. Later, we show that if a code $\mathcal{C}$ with $|\mathcal{C}|>3$ has its CCG to be connected 2-regular then $|\mathcal{C}| $ is even. The GRG was defined only for degree two neural codes using the canonical forms of its neural ideal. We first define GRG for any neural code. Then, we show the behaviour of GRGs under the various elementary code maps. At last, we compare these two graphs for certain classes of codes and see their properties.