171,914 research outputs found
A Sum of Squares Characterization of Perfect Graphs
We present an algebraic characterization of perfect graphs, i.e., graphs for
which the clique number and the chromatic number coincide for every induced
subgraph. We show that a graph is perfect if and only if certain nonnegative
polynomials associated with the graph are sums of squares. As a byproduct, we
obtain several infinite families of nonnegative polynomials that are not sums
of squares through graph-theoretic constructions. We also characterize graphs
for which the associated polynomials belong to certain structured subsets of
sum of squares polynomials. Finally, we reformulate some well-known results
from the theory of perfect graphs as statements about sum of squares proofs of
nonnegativity of certain polynomials
Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes
We prove that there exist bipartite Ramanujan graphs of every degree and
every number of vertices. The proof is based on analyzing the expected
characteristic polynomial of a union of random perfect matchings, and involves
three ingredients: (1) a formula for the expected characteristic polynomial of
the sum of a regular graph with a random permutation of another regular graph,
(2) a proof that this expected polynomial is real rooted and that the family of
polynomials considered in this sum is an interlacing family, and (3) strong
bounds on the roots of the expected characteristic polynomial of a union of
random perfect matchings, established using the framework of finite free
convolutions we recently introduced
On (Subgame Perfect) Secure Equilibrium in Quantitative Reachability Games
We study turn-based quantitative multiplayer non zero-sum games played on
finite graphs with reachability objectives. In such games, each player aims at
reaching his own goal set of states as soon as possible. A previous work on
this model showed that Nash equilibria (resp. secure equilibria) are guaranteed
to exist in the multiplayer (resp. two-player) case. The existence of secure
equilibria in the multiplayer case remained and is still an open problem. In
this paper, we focus our study on the concept of subgame perfect equilibrium, a
refinement of Nash equilibrium well-suited in the framework of games played on
graphs. We also introduce the new concept of subgame perfect secure
equilibrium. We prove the existence of subgame perfect equilibria (resp.
subgame perfect secure equilibria) in multiplayer (resp. two-player)
quantitative reachability games. Moreover, we provide an algorithm deciding the
existence of secure equilibria in the multiplayer case.Comment: 32 pages. Full version of the FoSSaCS 2012 proceedings pape
Graphical condensation of plane graphs: a combinatorial approach
The method of graphical vertex-condensation for enumerating perfect matchings
of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003),
267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57)
and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this
paper, by a purely combinatorial method some explicit identities on graphical
vertex-condensation for enumerating perfect matchings of plane graphs (which do
not need to be bipartite) are obtained. As applications of our results, some
results on graphical edge-condensation for enumerating perfect matchings are
proved, and we count the sum of weights of perfect matchings of weighted Aztec
diamond.Comment: 13 pages, 5 figures. accepted by Theoretial Computer Scienc
On subgroup perfect codes in Cayley sum graphs
A perfect code in a graph is an independent set of vertices of
such that every vertex outside of is adjacent to a unique vertex
in , and a total perfect code in is a set of vertices of
such that every vertex of is adjacent to a unique vertex in
. Let be a finite group and a normal subset of . The Cayley sum
graph of with the connection set is the graph with
vertex set and two vertices and being adjacent if and only if
and . In this paper, we give some necessary conditions of a
subgroup of a given group being a (total) perfect code in a Cayley sum graph of
the group. As applications, the Cayley sum graphs of some families of groups
which admit a subgroup as a (total) perfect code are classified
Perfect state transfer in cubelike graphs
Suppose is a subset of non-zero vectors from the vector space
. The cubelike graph has as its vertex
set, and two elements of are adjacent if their difference is
in . If is the matrix with the elements of as its
columns, we call the row space of the code of . We use this code to
study perfect state transfer on cubelike graphs. Bernasconi et al have shown
that perfect state transfer occurs on at time if and only if the
sum of the elements of is not zero. Here we consider what happens when this
sum is zero. We prove that if perfect state transfer occurs on a cubelike
graph, then it must take place at time , where is the greatest
common divisor of the weights of the code words. We show that perfect state
transfer occurs at time if and only if D=2 and the code is
self-orthogonal.Comment: 10 pages, minor revision
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