211,376 research outputs found
Sequence mixed graphs
A mixed graph can be seen as a type of digraph containing some edges (or two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and literated line digraphs. These structures are proven to be useful in the problem of constructing dense graphs or digraphs, and this is related to the degree/diameter problem. Thus, our generalized approach gives rise to graphs that have also good ratio order/diameter. Moreover, we propose a general method for obtaining a sequence mixed diagraph by identifying some vertices of certain iterated line digraph. As a consequence, some results about distance-related parameters (mainly, the diameter and the average distance) of sequence mixed graphs are presented.Postprint (author's final draft
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
Spectrum of mixed bi-uniform hypergraphs
A mixed hypergraph is a triple , where is
a set of vertices, and are sets of hyperedges. A
vertex-coloring of is proper if -edges are not totally multicolored and
-edges are not monochromatic. The feasible set of is the set of
all integers, , such that has a proper coloring with colors.
Bujt\'as and Tuza [Graphs and Combinatorics 24 (2008), 1--12] gave a
characterization of feasible sets for mixed hypergraphs with all - and
-edges of the same size , .
In this note, we give a short proof of a complete characterization of all
possible feasible sets for mixed hypergraphs with all -edges of size
and all -edges of size , where . Moreover, we show that
for every sequence , , of natural numbers there
exists such a hypergraph with exactly proper colorings using colors,
, and no proper coloring with more than colors. Choosing
this answers a question of Bujt\'as and Tuza, and generalizes
their result with a shorter proof.Comment: 9 pages, 5 figure
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
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
Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution
Even though power-law or close-to-power-law degree distributions are
ubiquitously observed in a great variety of large real networks, the
mathematically satisfactory treatment of random power-law graphs satisfying
basic statistical requirements of realism is still lacking. These requirements
are: sparsity, exchangeability, projectivity, and unbiasedness. The last
requirement states that entropy of the graph ensemble must be maximized under
the degree distribution constraints. Here we prove that the hypersoft
configuration model (HSCM), belonging to the class of random graphs with latent
hyperparameters, also known as inhomogeneous random graphs or -random
graphs, is an ensemble of random power-law graphs that are sparse, unbiased,
and either exchangeable or projective. The proof of their unbiasedness relies
on generalized graphons, and on mapping the problem of maximization of the
normalized Gibbs entropy of a random graph ensemble, to the graphon entropy
maximization problem, showing that the two entropies converge to each other in
the large-graph limit
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