10,094 research outputs found

    Inverse monoids of partial graph automorphisms

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    A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well

    Spanners for Geometric Intersection Graphs

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    Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.Comment: 16 pages, 5 figures, Late

    Ramanujan Coverings of Graphs

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    Let GG be a finite connected graph, and let ρ\rho be the spectral radius of its universal cover. For example, if GG is kk-regular then ρ=2k1\rho=2\sqrt{k-1}. We show that for every rr, there is an rr-covering (a.k.a. an rr-lift) of GG where all the new eigenvalues are bounded from above by ρ\rho. It follows that a bipartite Ramanujan graph has a Ramanujan rr-covering for every rr. This generalizes the r=2r=2 case due to Marcus, Spielman and Srivastava (2013). Every rr-covering of GG corresponds to a labeling of the edges of GG by elements of the symmetric group SrS_{r}. We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava, a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from a recent paper of them (2015). Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the rr-th matching polynomial of GG to be the average matching polynomial of all rr-coverings of GG. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside [ρ,ρ]\left[-\rho,\rho\right].Comment: 38 pages, 4 figures, journal version (minor changes from previous arXiv version). Shortened version appeared in STOC 201
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