We show that the number of points with pairwise different sets of neighbors in a graph is O(2) where r is the rank of the adjacency matrix. We also give an example that achieves this bound.

Let G be a graph with two non adjacent vertices and G′ the graph constructed from G by adding an edge between them. It is known that the trace of Q′ is 2 plus the trace of Q, where Q and Q′ are the signless Laplacian matrices of G and G′ respectively. So, the sum of the Q′-eigenvalues of G′ is the sum of the the Qeigenvalues of G plus two. It is said that Q-spectral integral variation occurs wh...

In 2012 Seager introduced a new variant of the Cops and Robbers game, in which a cop searches for a moving robber on a graph using distance probes. The robber on his turn can either remain still or move to an adjacent vertex, while the cop on her turn probes any vertex and learns the robber’s distance from the probed vertex. Carraher, Choi, Delcourt, Erickson and West later showed that for any ...

Residents of socioeconomically deprived areas perceive their neighbourhood as less conducive to healthy behaviours than residents of more affluent areas. Whether these unfavourable perceptions are based on objective neighbourhood features or other factors is poorly understood. We examined individual and contextual correlates of socioeconomic inequalities in neighbourhood perceptions across five...

In graph pebbling games, one considers a distribution of pebbles on the vertices of a graph, and a pebbling move consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The t-pebbling number πt(G) of a graph G is the smallest m such that for every initial distribution of m pebbles on V (G) and every target vertex x there exists a sequence of pebbling moves leading t...

We consider a variation of cop vs. robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on V (G) and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We show that when the gambler’s distribution...

A graph G is given and two players, a cop and a robber, play the folioking game: the cop chooses a vertex, then the robber chooses a vertex, then the players move alternately beginning with the cop. A move consists of staying at one’s present vertex or moving to an adjacent vertex; each move is seen by both players. The cop wins if he manages to occupy the same vertex as the robber, and the rob...

We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that π(G × H) ≤ π(G)π(H), where π(G) is the pebbling number of the graph G. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that show that Sjöstrand’s theorem on cover p...

Neighbourhood Effects Research at a Crossroads: Ten Challenges for Future Research Neighbourhood effects research is at a crossroads since current theoretical and empirical approaches do not seem to be moving the debate forward. In this paper, we present a set of ten challenges as a basis for a new research agenda which will give new direction to the neighbourhood effects debate. The ten challe...

Given a polygon P in the plane, a pop operation is the reflection of a vertex with respect to the line through its adjacent vertices. We define a family of alternating polygons, and show that any polygon from this family cannot be convexified by pop operations. This family contains simple, as well as non-simple (i.e., self-intersecting) polygons, as desired. We thereby answer in the negative an...