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 $i,j \in \lbrace 1, ... , m \rbrace$, every vertex of A_i is adjacent to the same number of vertices, namely, a_{ij} vertices, of A_j. The matrix $A=(a_{ij})_{i,j\in \lbrace 1,... ,m\rbrace }$, 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 $Z_2^{d_1} \times Z_4^{d_2}$. We further provide an infinite family of examples of cubelike graphs (Cayley graphs of $Z_2^d$ ) in which this set has size at least four, covering all possible values of $d$. 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 $G$ be a graph, $V (G)$ and $E(G)$ be the vertex set and edge set of $G$, respectively. A perfect matching of $G$ is a set of edges, $M\subseteq E(G)$, such that each vertex in $G$ is incident with exactly one edge in $M$. An $r$-regular graph is said to be an $r$-graph if $|\partial(X)| \geq r$ for each odd set $X \subseteq V(G)$, where $|\partial(X)|$ denotes the set of edges with precisely one end in $X$. One of the most famous conjectures in Matching Theory, due to Berge, states that every 3-graph $G$ has five perfect matchings such that each edge of $G$ is contained in at least one of them. Likewise, generalization of the Berge Conjecture given, by Seymour, asserts that every $r$-graph $G$ has $2r-1$ perfect matchings that covers each $e \in E(G)$ 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 $r$-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 $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that mean subtree order of $H$ is smaller then mean subtree order of $G$

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