Generalized hypercubes and (0,2)-graphs

AbstractA generalized hypercube Qd(S) (S ⊆ {1, 2, …, d}) has {0,1}d as vertex set and two vertices are joined whenever their mutual distance in Qd belongs to S. These graphs have been introduced in (Berrachedi and Mollard, 1996) where the notion mainly investigated there is graph embedding, especially, in the case where the host graph is a hypercube. A simple connected graph G is a (0, 2)-graph if any two vertices have 0 or exactly two common neighbors as introduced in (Mulder, 1980). We give first some results about the structure of generalized hypercubes, and then characterize those of which are (0, 2)-graphs. Using similar construction as in generalized hypercubes, we exhibit a class of (0, 2)-graphs which are not vertex transitive which contradicts again a conjecture of Mulder (1982) on the convexity of interval regular graphs

The induced path function, monotonicity and betweenness

The induced path function $J(u, v)$ of a graph consists of the set of all vertices lying on the induced paths between vertices $u$ and $v$. This function is a special instance of a transit function. The function $J$ satisfies betweenness if $w \\in J(u, v)$ implies $u \\notin J(w, v)$ and $x \\in J(u, v)$ implies $J(u, x \\subseteq J(u, v)$, and it is monotone if $x, y \\in J(u, v)$ implies $J(x, y) \\subseteq J(u, v)$. The induced path function of aconnected graph satisfying the betweenness and monotone axioms are characterized by transit axioms.betweenness;induced path;transit function;monotone;house domino;long cycle;p-graph

Consensus Strategies for Signed Profiles on Graphs

The median problem is a classical problem in Location Theory: one searches for a location that minimizes the average distance to the sites of the clients. This is for desired facilities as a distribution center for a set of warehouses. More recently, for obnoxious facilities, the antimedian was studied. Here one maximizes the average distance to the clients. In this paper the mixed case is studied. Clients are represented by a profile, which is a sequence of vertices with repetitions allowed. In a signed profile each element is provided with a sign from {+,-}. Thus one can take into account whether the client prefers the facility (with a + sign) or rejects it (with a - sign). The graphs for which all median sets, or all antimedian sets, are connected are characterized. Various consensus strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity. Hypercubes are the only graphs on which Majority produces the median set for all signed profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming graphs, Johnson graphs and halfcubes.median;consensus function;median graph;majority rule;plurality strategy;Graph theory;Hamming graph;Johnson graph;halfcube;scarcity strategy;Discrete location and assignment;Distance in graphs

A characterization of bipartite graphs associated with BIB-designs with λ = 1

AbstractA graph is said to be F-geodetic (for some function F) if the number of shortest paths between two vertices at distance i is F(i). It is shown that a bipartite F-geodetic graph with diameter ⩽4 is either (i)a tree, or(ii)a distance-regular graph, or(iii)the graph associated with a BIB-design with λ = 1