424 research outputs found
Nonlinear spectral calculus and super-expanders
Nonlinear spectral gaps with respect to uniformly convex normed spaces are
shown to satisfy a spectral calculus inequality that establishes their decay
along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to
behave sub-multiplicatively under zigzag products. These results yield a
combinatorial construction of super-expanders, i.e., a sequence of 3-regular
graphs that does not admit a coarse embedding into any uniformly convex normed
space.Comment: Typos fixed based on referee comments. Some of the results of this
paper were announced in arXiv:0910.2041. The corresponding parts of
arXiv:0910.2041 are subsumed by the current pape
On the geodesic pre-hull number of a graph
AbstractGiven a convexity space X whose structure is induced by an interval operator I, we define a parameter, called the pre-hull number of X, which measures the intrinsic non-convexity of X in terms of the number of iterations of the pre-hull operator associated with I which are necessary in the worst case to reach the canonical extension of copoints of X when they are being extended by the adjunction of an attaching point. We consider primarily the geodesic convexity structure of connected graphs in the case where the pre-hull number is at most 1, with emphasis on bipartite graphs, in particular, partial cubes
On a generalization of median graphs: -median graphs
Median graphs are connected graphs in which for all three vertices there is a
unique vertex that belongs to shortest paths between each pair of these three
vertices. To be more formal, a graph is a median graph if, for all , it holds that where
denotes the set of all vertices that lie on shortest paths connecting
and . In this paper we are interested in a natural generalization of
median graphs, called -median graphs. A graph is a -median graph, if
there are vertices such that, for all , it holds that , . By definition, every median graph with vertices is an -median graph.
We provide several characterizations of -median graphs that, in turn, are
used to provide many novel characterizations of median graphs
Partial cubes: structures, characterizations, and constructions
AbstractPartial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djoković’s and Winkler’s relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given.The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated
Finding Connected Dense -Subgraphs
Given a connected graph on vertices and a positive integer ,
a subgraph of on vertices is called a -subgraph in . We design
combinatorial approximation algorithms for finding a connected -subgraph in
such that its density is at least a factor
of the density of the densest -subgraph
in (which is not necessarily connected). These particularly provide the
first non-trivial approximations for the densest connected -subgraph problem
on general graphs
Finding a Maximum 2-Matching Excluding Prescribed Cycles in Bipartite Graphs
We introduce a new framework of restricted 2-matchings close to Hamilton cycles. For an undirected graph (V,E) and a family U of vertex subsets, a 2-matching F is called U-feasible if, for each setU in U, F contains at most |setU|-1 edges in the subgraph induced by U. Our framework includes C_{= 5. For instance, in bipartite graphs in which every cycle of length six has at least two chords, our algorithm solves the maximum C_{<=6}-free 2-matching problem in O(n^2 m) time, where n and m are the numbers of vertices and edges, respectively
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