An infinite family of tetravalent half-arc-transitive graphs

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. In this paper, a new infinite family of tetravalent half-arc-transitive graphs with girth 4 is constructed, each of which has order 16m16m such that m>1m>1 is a divisor of 2t2+2t+12t2+2t+1 for a positive integer t   and is tightly attached with attachment number 4m4m. The smallest graph in the family has order 80.