In this thesis we classify primitive amalgams of degree (5,2) and thereby obtain a description of symmetric graphs of valency 5. There are 22 such amalgams, and together with this list, we find the presentations for the universal completions of such amalgams and an example of a faithful finite completion in each case. Our methods include group theory, the amalgam method and some modular represe...

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, we classify the s-regular elementary Abelian coverings of the three-dimensional hypercube for each s ≥ 1 whose fibre-preserving automorphism subgroups act arc-transitively. This gives a new infinite family of cubic 1-regular graphs, in which the smallest one has order 19 208. As an application...

Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs, and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of th...

In this paper we examine spectral properties of random intersection graphs when the number of vertices is equal to the number of labels. We call this class symmetric random intersection graphs. We examine symmetric random intersection graphs when the probability that a vertex selects a label is close to the connectivity threshold τc. In particular, we examine the size of the second eigenvalue o...

A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary gr...

We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means t...

We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges its two parts. To prove this, we study the relationship between symmetric bipartite graphs and graphs with loops.

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, a complete classification of connected pentavalent symmetric graphs of order 12p is given for each prime p. As a result, a connected pentavalent symmetric graph of order 12p exists if and only if p = 2, 3, 5 or 11, and up to isomorphism, there are only nine such graphs: one for each p = 2, 3 ...

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. We show that a symmetric cubic graph of girth at most 9 is either 1-regular or 2-regular (following the notation of Djokovic), or...

Abstract We characterize seven valent symmetric graphs of order 2 p q n 2p{q}^{n} with < p\lt q odd primes, extending a few previous results. Moreover, consequence partially generalizes the result Conder, Li and Potočnik [ On o...