Exact Counts of $C_{4}$s in Blow-Up Graphs

Abstract

Cycles have many interesting properties and are widely studied in many disciplines. In some areas, maximising the counts of $k$-cycles are of particular interest. A natural candidate for the construction method used to maximise the number of subgraphs $H$ in a graph $G$, is the \emph{blow-up} method. Take a graph $G$ on $n$ vertices and a pattern graph $H$ on $k$ vertices, such that $n\geq k$, the blow-up method involves an iterative process of replacing vertices in $G$ with a copy of the $k$-vertex graph $H$. In this paper, we apply the blow-up method on the family of cycles. We then present the exact counts of cycles of length 4 for using this blow-up method on cycles and generalised theta graphs.