ARITHMETICAL STRUCTURES ON

Gromov Hyperbolicity of Substitution graphs

In this paper, we construct a class of infinite graphs, called substitution graphs. The vertex set consists of all finite words over a finite alphabet. A directed graph is formed by adding vertical edges connecting each word to its children and horizontal edges defined recursively by two finite directed graphs G and J:...

Pendant paths and integral generalized sun graphs

A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. We prove that every simple graph having a pendant path with at least three edges has an eigenvalue in $(1,2\cos(\pi/9)]$ and one in $[-2\cos(\pi/9),-1)$, and hence is not integral. This settles a conjecture of Braga, Del-Vecchio...

A finiteness theorem for geodesic Leech wheels

Let f be a labeling of the edges of a finite graph G by positive integers, and let the weight of a path be the sum of the labels of its edges. The labeling is a geodesic Leech labeling if the weights of the geodesics are exactly 1, 2, ..., t_gp(G), each occurring once, where t_gp(G) is the geodesic path number of G. Le...

A construction of F-irregular graphs

For a fixed graph F, the F-degree of a vertex v in a host graph H is the number of subgraphs of H isomorphic to F that contain v, and H is F-irregular if its F-degrees are pairwise distinct. We show that every finite connected graph F on at least three vertices admits a finite connected F-irregular host. For noncomplet...

Maximal Hamiltonicity of realization graphs of degree sequences

We prove that the realization graph of every graphical degree sequence is maximally Hamiltonian: it is Hamilton-laceable when bipartite on more than one vertex, and Hamilton-connected otherwise. This answers Problem P59 of M\"utze's survey of combinatorial Gray codes, and the Hamiltonicity question recorded as open by...

Rose-Hulman Undergraduate Mathematics