129,011 research outputs found
Spectrum of Markov generators on sparse random graphs
Correction in Proposition 4.3. Final version.International audienceWe investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if kj and L(j,j)=-sum(L(j,k),kj), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented ErdĆs-RĂ©nyi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L
On semi-transitive orientability of Kneser graphs and their complements
An orientation of a graph is semi-transitive if it is acyclic, and for any
directed path either
there is no edge between and , or is an edge
for all . An undirected graph is semi-transitive if it admits
a semi-transitive orientation. Semi-transitive graphs include several important
classes of graphs such as 3-colorable graphs, comparability graphs, and circle
graphs, and they are precisely the class of word-representable graphs studied
extensively in the literature.
In this paper, we study semi-transitive orientability of the celebrated
Kneser graph , which is the graph whose vertices correspond to the
-element subsets of a set of elements, and where two vertices are
adjacent if and only if the two corresponding sets are disjoint. We show that
for , is not semi-transitive, while for , is semi-transitive. Also, we show computationally that a
subgraph on 16 vertices and 36 edges of , and thus itself
on 56 vertices and 280 edges, is non-semi-transitive. and are the
first explicit examples of triangle-free non-semi-transitive graphs, whose
existence was established via Erd\H{o}s' theorem by Halld\'{o}rsson et al. in
2011. Moreover, we show that the complement graph of
is semi-transitive if and only if
Embedding problems in graphs and hypergraphs
In this thesis, we explore several mathematical questions about substructures in graphs and hypergraphs, focusing on algorithmic methods and notions of regularity for graphs and hypergraphs. We investigate conditions for a graph to contain powers of paths and cycles of arbitrary specified linear lengths. Using the well-established graph regularity method, we determine precise minimum degree thresholds for sufficiently large graphs and show that the extremal behaviour is governed by a family of explicitly given extremal graphs. This extends an analogous result of Allen, Böttcher and HladkĂœ for squares of paths and cycles of arbitrary specified linear lengths and confirms a conjecture of theirs. Given positive integers k and j with j < k, we study the length of the longest j-tight path in the binomial random k-uniform hypergraph Hk(n, p). We show that this length undergoes a phase transition from logarithmic to linear and determine the critical threshold for this phase transition. We also prove upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the Pathfinder algorithm, a depth-first search algorithm which discovers j-tight paths in a k-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm finds a long j-tight path. Finally, we investigate the embedding of bounded degree hypergraphs into large sparse hypergraphs. The blow-up lemma is a powerful tool for embedding bounded degree spanning subgraphs with wide-ranging applications in extremal graph theory. We prove a sparse hypergraph analogue of the blow-up lemma, showing that large sparse partite complexes with sufficiently regular small subcomplex counts and no atypical vertices behave as if they were complete for the purpose of embedding complexes with bounded degree and bounded partite structure
A Linear Kernel for Planar Total Dominating Set
A total dominating set of a graph is a subset such
that every vertex in is adjacent to some vertex in . Finding a total
dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on
general graphs when parameterized by the solution size. By the meta-theorem of
Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total
Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how
such a kernel can be effectively constructed, and how to obtain explicit
reduction rules with reasonably small constants. Following the approach of
Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating
Set on planar graphs with at most vertices, where is the size of the
solution. This result complements several known constructive linear kernels on
planar graphs for other domination problems such as Dominating Set, Edge
Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue
Dominating Set.Comment: 33 pages, 13 figure
Asymptotic enumeration of dense 0-1 matrices with specified line sums
Let S=(s_1,s_2,..., s_m) and T = (t_1,t_2,..., t_n) be vectors of
non-negative integers with sum_{i=1}^{m} s_i = sum_{j=1}^n t_j. Let B(S,T) be
the number of m*n matrices over {0,1} with j-th row sum equal to s_j for 1 <= j
<= m and k-th column sum equal to t_k for 1 <= k <= n. Equivalently, B(S,T) is
the number of bipartite graphs with m vertices in one part with degrees given
by S, and n vertices in the other part with degrees given by T. Most research
on the asymptotics of B(S,T) has focused on the sparse case, where the best
result is that of Greenhill, McKay and Wang (2006). In the case of dense
matrices, the only precise result is for the case of equal row sums and equal
column sums (Canfield and McKay, 2005). This paper extends the analytic methods
used by the latter paper to the case where the row and column sums can vary
within certain limits. Interestingly, the result can be expressed by the same
formula which holds in the sparse case.Comment: Multiple minor adjustments. Accepted by JCT-
OV Graphs Are (Probably) Hard Instances
© Josh Alman and Virginia Vassilevska Williams. A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v1, . . ., vn â {0, 1}d such that nodes i and j are adjacent in G if and only if hvi, vji = 0 over Z. In this paper, we study a number of basic graph algorithm problems, except where one is given as input the vectors defining an OV graph instead of a general graph. We show that for each of the following problems, an algorithm solving it faster on such OV graphs G of dimension only d = O(log n) than in the general case would refute a plausible conjecture about the time required to solve sparse MAX-k-SAT instances: Determining whether G contains a triangle. More generally, determining whether G contains a directed k-cycle for any k â„ 3. Computing the square of the adjacency matrix of G over Z or F2. Maintaining the shortest distance between two fixed nodes of G, or whether G has a perfect matching, when G is a dynamically updating OV graph. We also prove some complementary results about OV graphs. We show that any problem which is NP-hard on constant-degree graphs is also NP-hard on OV graphs of dimension O(log n), and we give two problems which can be solved faster on OV graphs than in general: Maximum Clique, and Online Matrix-Vector Multiplication
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