Symmetric graphs, regular maps and their central covers

Abstract

In this thesis, we investigate symmetric graphs, regular maps and their central covers.

Let $\Gamma=(V,E)$ be a $G$-vertex transitive graph and $\mathcal B$ a $G$-invariant partition of $V$. The imprimitive quotient $\Gamma_{\mathcal{B}}$ is the graph with vertex $\mathcal{B}$ such that a vertex $B$ is connected to a vertex $B'$ if and only if a vertex $v\in B$ in $\Gamma$ is connected to a vertex $v'\in B'$ in $\Gamma$. For two adjacent vertices $B,B'\in \mathcal{B}$ in $\Gamma_{\mathcal{B}}$, if the induced subgraph $[B,B']$ of $\Gamma$ is a perfect matching, $\Gamma$ is called a cover, or covering of $\Gamma_{\mathcal B}$. The investigation of covering graph for symmetric graphs began with a construction of highly symmetric cubic graphs by Conway \cite{Biggs}. Since then, the study of covers of graphs has received much attention in the literature. In this thesis, we characterize the central cover for a symmetric graph. Let $\Gamma=(V,E)$ be a connected graph, and let $X\leq \Aut(\Gamma)$ be edge transitive. Let $N\lhd X$ be a normal subgroup of $X$. We can define a normal quotient graph $\Gamma_N$ as follows: the vertex set is the orbits of $N$ on $V$ with adjacent relation: two orbits $V_1$ and $V_2$ are adjacent if and only if $v_1$ is adjacent to $v_2$ in $\Gamma$ for some $v_1\in V_1$ and $v_2\in V_2$. Let $\Sigma=\Gamma_N$ be the normal quotient graph. Assume $\Gamma$ is a cover of $\Sigma$. Then we call $\Gamma$ a central cover of $\Sigma$ if the induced action of $X$ on $N$ by conjugation is the inner automorphism group of $N$ acting on $N$, that is, $$\bfN_X(N)/\bfC_X(N)=X/\bfC_X(N)\cong N.$$ We mainly give the characterization of connected locally primitive graphs which are central covers of some locally primitive graphs. We divide the characterization into five cases by the action of $\bfC_X(N)$ on $\Gamma$. In each case, we give the characterization for the central cover and construct various examples. Also, we characterize minimal normal covers of arc-transitive graphs which are vertex-quasiprimitive and locally-primitive.

A map is regular if the automorphism group is transitive on the flag set. A regular map can be regarded as a coset map which is flag regular. Edge transitive maps are categorized into fourteen classes according to local structures and local actions, in \cite{GW,ST}. The problem of constructing and classifying special classes of such maps with specific Euler characteristic has attracted considerable attention. In this thesis, we investigate automorphism groups of a regular map with special Euler characteristic. First, we investigate the finite groups satisfying: each Sylow $p$-subgroup has a cyclic or dihedral subgroup of index $p$, as we restrict the square free Euler characteristic. Second, the automorphism group of a regular map is generated by three involutions with two of which commuting. Combining these two conditions together, we give our characterizations of finite groups acting on the regular maps with square free Euler characteristic, and we also construct many regular maps satisfying the condition of the required Euler characteristic.