On tetravalent half-arc-transitive graphs of girth 5

A subgroup of the automorphism group of a graph $\text{G}$ is said to be {\em half-arc-transitive} on $\text{G}$ if its action on $\text{G}$ is transitive on the vertex set of $\text{G}$ and on the edge set of $\text{G}$ but not on the arc set of $\text{G}$. Tetravalent graphs of girths $3$ and $4$ admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth $5$. We show that, with two exceptions, all such graphs only have directed $5$-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed $5$-cycles, study some of their graph theoretic properties and prove that the $5$-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth $5$ admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth $5$.