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8,203
Property Testing in Bounded Degree Graphs
 Algorithmica
, 1997
"... We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength in ..."
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Cited by 124 (36 self)
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length incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of boundeddegree graphs. In particular, we present randomized algorithms
Degree Graphs of Simple Groups
 Rocky Mountain J. of Math
"... Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph ∆(G) is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). We compile here the graphs ∆(G) ..."
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Cited by 2 (1 self)
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Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph ∆(G) is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). We compile here the graphs ∆(G
On testing expansion in boundeddegree graphs
 Electronic Colloquium on Computational Complexity (ECCC
, 2000
"... Abstract. We consider testing graph expansion in the boundeddegree graph model. Specifically, we refer to algorithms for testing whether the graph has a second eigenvalue bounded above by a given threshold or is far from any graph with such (or related) property. We present a natural algorithm aime ..."
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Cited by 67 (5 self)
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Abstract. We consider testing graph expansion in the boundeddegree graph model. Specifically, we refer to algorithms for testing whether the graph has a second eigenvalue bounded above by a given threshold or is far from any graph with such (or related) property. We present a natural algorithm
Search on High Degree Graphs
 IN 17TH INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 2001
"... We show that nodes of high degree tend to occur infrequently in random graphs but frequently in a wide variety of graphs associated with real world search problems. We then study some alternative models for randomly generating graphs which have been proposed to give more realistic topologies. ..."
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Cited by 23 (5 self)
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We show that nodes of high degree tend to occur infrequently in random graphs but frequently in a wide variety of graphs associated with real world search problems. We then study some alternative models for randomly generating graphs which have been proposed to give more realistic topologies
A Critical Point For Random Graphs With A Given Degree Sequence
, 2000
"... Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0 the ..."
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Cited by 507 (8 self)
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Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0
Testing expansion in boundeddegree graphs
 Proc. of FOCS 2007
"... We consider the problem of testing expansion in bounded degree graphs. We focus on the notion of vertexexpansion: an αexpander is a graph G = (V, E) in which every subset U ⊆ V of at most V /2 vertices has a neighborhood of size at least α · U. Our main result is that one can distinguish good ..."
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Cited by 15 (1 self)
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We consider the problem of testing expansion in bounded degree graphs. We focus on the notion of vertexexpansion: an αexpander is a graph G = (V, E) in which every subset U ⊆ V of at most V /2 vertices has a neighborhood of size at least α · U. Our main result is that one can distinguish good
Algebraic Graph Theory
, 2011
"... Algebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example, automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects. One of the oldest themes in the area is the investiga ..."
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Cited by 892 (13 self)
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Algebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example, automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects. One of the oldest themes in the area
Character degree graphs that are complete graphs
 Proc. AMS 135
, 2007
"... Abstract. Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define Γ(G) to be the graph whose vertex set is cd(G) −{1}, and there is an edge between a and b if (a, b)> 1. We prove that if Γ(G) is a complete graph, then G is a solvable group. 1. ..."
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Cited by 12 (2 self)
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Abstract. Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define Γ(G) to be the graph whose vertex set is cd(G) −{1}, and there is an edge between a and b if (a, b)> 1. We prove that if Γ(G) is a complete graph, then G is a solvable group. 1.
On Testing Expansion in BoundedDegree Graphs
 Electronic Colloquium on Computational Complexity
, 2000
"... We consider testing graph expansion in the boundeddegree graph model (as formulated in [1]). Specifically, we refer to algorithms for testing whether the graph has a second eigenvalue bounded above by a given threshold or is far from any graph with such (or related) property. We present a natural a ..."
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We consider testing graph expansion in the boundeddegree graph model (as formulated in [1]). Specifically, we refer to algorithms for testing whether the graph has a second eigenvalue bounded above by a given threshold or is far from any graph with such (or related) property. We present a natural
Separation dimension of bounded degree graphs
"... The separation dimension of a graph G is the smallest natural number k for which the vertices of G can be embedded in Rk such that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family F of total ..."
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degree graphs and show that the separation dimension of a graph with maximum degree d is at most 29 log? dd. We also demonstrate that the above bound is nearly tight by showing that, for every d, almost all dregular graphs have separation dimension at least dd/2e.
Results 1  10
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8,203