Paper detail

Isotemporal classes of diasters, beachballs, and daisies

If the vertices composing a network interact at distinct time points, the temporal ordering of these interactions and the network's graph structure are sufficient to convey the routes by which information can flow in the network. Two networks with real-valued edge labels are temporally isomorphic if there exists a graph isomorphism f:N->M that preserves temporal paths - paths in which sequential edge labels are strictly increasing. An equivalence class of temporally ismorphic networks is known as an isotemporal class. Methods to determine the number of isotemporal classes of a particular graph structure N(G) are non-obvious, and refractory to traditional techniques such as Pólya enumeration (Pólya, 1937). Here, I present a simple formula for the number of isotemporal classes of diasters, graphs composed of a vertex of degree a+1 connected to a vertex of degree b+1, with all other vertices of degree 1 (denoted D(a,b)). In particular, N(D(a,b)))=ab+a+b+1 if a is not equal to b, and N(D(a,a)))=(1/2) (a^2+3a+2) otherwise. This formula is then extended to five additional types of pseudograph by application of a theorem that states N(G) is preserved between two graph types if edge adjacencies and automorphisms are preserved, and provided that any two networks are members of the same isotemporal class if and only if they are isomorphic by transpositions of sequential edge labels on non-adjacent edges.