The edge-girth-regularity of Wenger graphs

Let $n\ge 1$ be an integer and $\mathbb{F}_q$ be a finite field of characteristic $p$ with $q$ elements. In this paper, it is proved that the Wenger graph $W_n(q)$ and linearized Wenger graph $L_m(q)$ are edge-girth-regular $(v,k,g,\lambda)$-graphs, and the parameter $\lambda$ of graphs $W_n(q)$ and $L_m(q)$ is completely determined. Here, an edge-girth-regular graph $egr(v,k,g,\lambda)$ means a $k$-regular graph of order $v$ and girth $g$ satisfying that any edge is contained in $\lambda$ distinct $g$-cycles. As a direct corollary, we obtain the number of girth cycles of graph $W_n(q)$, and the lower bounds on the generalized Tur\'an numbers $ex(n, C_{6}, \mathscr{C}_{5})$ and $ex(n, C_{8}, \mathscr{C}_{7})$, where $C_k$ is the cycle of length $k$ and $\mathscr{C}_k = \{C_3, C_4, \dots , C_k\}$.Moreover, there exist a family of $egr(2q^3,q,8,(q-1)^3(q-2))$-graphs for $q$ odd, and the order of graph $W_2(q)$ and extremal $egr(v,q,8,(q-1)^3(q-2))$-graph have same asymptotic order for $q$ odd.