If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha $ is an automorphism of X, then $\alpha ({\mathbf v})$ is also an eigenvector for $\lambda $. Thus, it is rather exceptional for an eigenvalue of a vertex-transitive graph to have multiplicity one. We study cubic vertex-transitive graphs with a nontrivial simple eigenvalue, and discover remarkable connections to arc-transitivity, regular maps, and number theory.

Graph embedding is a transformation of nodes of a network into a set of vectors. A good embedding should capture the underlying graph topology and structure, node-to-node relationship, and other relevant information about the graph, its subgraphs, and nodes themselves. If these objectives are achieved, an embedding is a meaningful, understandable, and often compressed representation of a network. Unfortunately, selecting the best embedding is a challenging task and very often requires domain experts. In this paper, we extend the framework for evaluating graph embeddings that was recently introduced in [15]. Now, the framework assigns two scores, local and global, to each embedding that measure the quality of an evaluated embedding for tasks that require good representation of local and, respectively, global properties of the network. The best embedding, if needed, can be selected in an unsupervised way, or the framework can identify a few embeddings that are worth further investigation. The framework is flexible and scalable and can deal with undirected/directed and weighted/unweighted graphs.

We consider cyclic graphs, that is, graphs with cyclic orders at the vertices, corresponding to 2-cell embeddings of graphs into orientable surfaces, or combinatorial maps. We construct a three variable polynomial invariant of these objects, the cyclic graph polynomial, which has many of the useful properties of the Tutte polynomial. Although the cyclic graph polynomial generalizes the Tutte polynomial, its definition is very different, and it depends on the embedding in an essential way. 2000 Mathematical Subject Classification: 05C10.