52,566 research outputs found
Perfect 3-colorings of the Cubic Graphs of Order 10
Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte. A perfect m-coloring of a graph G with m colors is a partition of the vertex set of G into m parts A_1, A_2, ..., A_m such that, for all , every vertex of A_i is adjacent to the same number of vertices, namely, a_{ij} vertices, of A_j. The matrix , is called the parameter matrix. We study the perfect 3-colorings (also known as the equitable partitions into three parts) of the cubic graphs of order 10. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the cubic graphs of order 10
State Transfer & Strong Cospectrality in Cayley Graphs
This thesis is a study of two graph properties that arise from quantum walks: strong cospectrality of vertices and perfect state transfer. We prove various results about these properties in Cayley graphs.
We consider how big a set of pairwise strongly cospectral vertices can be in a graph. We prove an upper bound on the size of such a set in normal Cayley graphs in terms of the multiplicities of the eigenvalues of the graph. We then use this to prove an explicit bound in cubelike graphs and more generally, Cayley graphs of . We further provide an infinite family of examples of cubelike graphs (Cayley graphs of ) in which this set has size at least four, covering all possible values of .
We then look at perfect state transfer in Cayley graphs of abelian groups having a cyclic Sylow-2-subgroup. Given such a group, G, we provide a complete characterization of connection sets C such that the corresponding Cayley graph for G admits perfect state transfer. This is a generalization of a theorem of Ba\v{s}i\'{c} from 2013, where he proved a similar characterization for
Cayley graphs of cyclic groups
Berge - Fulkerson Conjecture And Mean Subtree Order
Let be a graph, and be the vertex set and edge set of , respectively. A perfect matching of is a set of edges, , such that each vertex in is incident with exactly one edge in . An -regular graph is said to be an -graph if for each odd set , where denotes the set of edges with precisely one end in . One of the most famous conjectures in Matching Theory, due to Berge, states that every 3-graph has five perfect matchings such that each edge of is contained in at least one of them. Likewise, generalization of the Berge Conjecture given, by Seymour, asserts that every -graph has perfect matchings that covers each at least once. In the first part of this thesis, I will provide a lower bound to number of perfect matchings needed to cover the edge set of an -graph. I will also present some new conjectures that might shade a light towards the generalized Berge conjecture. In the second part, I will present a proof of a conjecture stating that there exists a pair of graphs and with , and such that mean subtree order of is smaller then mean subtree order of
Subsampled Power Iteration: a Unified Algorithm for Block Models and Planted CSP's
We present an algorithm for recovering planted solutions in two well-known
models, the stochastic block model and planted constraint satisfaction
problems, via a common generalization in terms of random bipartite graphs. Our
algorithm matches up to a constant factor the best-known bounds for the number
of edges (or constraints) needed for perfect recovery and its running time is
linear in the number of edges used. The time complexity is significantly better
than both spectral and SDP-based approaches.
The main contribution of the algorithm is in the case of unequal sizes in the
bipartition (corresponding to odd uniformity in the CSP). Here our algorithm
succeeds at a significantly lower density than the spectral approaches,
surpassing a barrier based on the spectral norm of a random matrix.
Other significant features of the algorithm and analysis include (i) the
critical use of power iteration with subsampling, which might be of independent
interest; its analysis requires keeping track of multiple norms of an evolving
solution (ii) it can be implemented statistically, i.e., with very limited
access to the input distribution (iii) the algorithm is extremely simple to
implement and runs in linear time, and thus is practical even for very large
instances
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