Narrow Results

Three disjoint rays in ℝ³ form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.

Yao and Theta graphs are defined for a given point set and a fixed integer k > 0. The space around each point is divided into k cones of equal angle, and each point is connected to a nearest neighbor in each cone. The difference between Yao and Theta graphs is in the way the nearest neighbor is defined: Yao graphs minimize the Euclidean distance between a point and its neighbor, and Theta graphs minimize the Euclidean distance between a point and the orthogonal projection of its neighbor on the bisector of the hosting cone. We prove that, corresponding to each edge of the Theta graph Θ₆, there is a path in the Yao graph Y₆ whose length is at most 8.82 times the edge length. Combined with the result of Bonichon et al., who prove an upper bound of 2 on the stretch factor of Θ₆, we obtain an upper bound of 17.64 on the stretch factor of Y₆.

Let $P_n$ and $C_n$ denote the path and cycle on $n$ vertices, respectively. The dumbbell graph, denoted by $D_{p,k,q}$, is the graph obtained from two cycles $C_p$, $C_q$ and a path $P_{k+2}$ by identifying each pendant vertex of $P_{k+2}$ with a vertex of a cycle, respectively. The theta graph, denoted by ${\mathrm{\Theta }}_{r,s,t}$, is the graph formed by joining two given vertices via three disjoint paths $P_r$, $P_s$ and $P_t$, respectively. In this paper, we prove that all dumbbell graphs as well as all theta graphs are determined by their $L$-spectra.