This chapter gives an overview of the problem of drawing graphs with as much symmetry as possible. It describes a linear time algorithm for constructing maximally symmetric straight-line drawings of planar graphs.

We present a simple polynomial-time algorithm that recognises reflexive, symmetric graphs admitting a near-unanimity operation. Several other characterisations of these graphs are also presented.  2004 Elsevier Inc. All rights reserved.

Let r be a simple undirected graph and G a subgroup of Aut r. r is said to be G-symmetric, if G acts transitively on the set of ordered adjacent pairs of vertices of r; r is said to be symmetric if it is Autrsymmetric. In this paper we give a complete classification for symmetric graphs of order 30. (See Theorem 10.)

We investigate subsets of the symmetric group with structure similar to that of a graph. The “trees” of these subsets then lead to minimal highly symmetric generating sets of the symmetric group. We show that there exist generating sets among these with edge-transitive Cayley graphs and investigate them in relation to the Lovasz conjecture.

We define a special sort of weighted oriented graphs, signed quivers. Each of these yields a symmetric quiver, i.e., a quiver endowed with an involutive anti-automorphism and the inherited signs. We develop a representation theory of symmetric quivers, in particular we describe the indecom-posable symmetric representations. Their dimensions constitute root systems corresponding to certain symme...

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We survey geometric properties which imply the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to the case of Riemannian manifolds, there exist examples of stochastically incomplete graphs of polynomial vo...

Spreads of finite symplectic, orthogonal and unitary vector spaces are used to construct new strongly regular graphs having the same parameters as the perpendicularity graphs of the underlying vector spaces. Some of the graphs are. related to partial geometries, while others produce interesting symmetric designs.

This paper describes the construction of all the vertex-transitive graphs on 24 vertices, thus extending the currently available catalogues. This construction differs significantly from previous constructions of the vertex-transitive graphs of order up to 23 in that we are forced to use far more sophisticated group-theoretic techniques. We include an analysis of all the symmetric graphs on 24 v...

We present symmetric and asymmetric similarity measures for labeled directed rooted graphs that are inspired by the simulation and bisimulation relations on labeled transition systems. Computation of the similarity measures has close connections to discounted Markov decision processes in the asymmetric case and to perfect-information stochastic games in the symmetric case. For the symmetric cas...