Several extremal problems on graphs involving the circumference, girth, and hyperbolicity constant

Abstract: To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let $\mathcal{G}(g,c,n)$ be the set of graphs $G$ with girth $g(G)=g$, circumference $c(G)=c$, and $n$ vertices; and let $\mathcal{H}(g,c,m)$ be the set of graphs with girth $g$, circumference $c$, and $m$ edges. In this work, we study the four following extremal problems on graphs: $A(g,c,n)=\min{\delta(G)\,|\; G \in \mathcal{G}(g,c,n) }$, $B(g,c,n)=\max{\delta(G)\,|\; G \in \mathcal{G}(g,c,n) }$, $\alpha(g,c,m)=\min{\delta(G)\,|\; \in \mathcal{H}(g,c,m) }$ and $\beta(g,c,m)=\max{\delta(G)\,|\; G \in \mathcal{H}(g,c,m) }$. In particular, we obtain bounds for $A(g,c,n)$ and $\alpha(g,c,m)$, and we compute the precise value of $B(g,c,n)$ and $\beta(g,c,m)$ for all values of $g$, $c$, $n$ and $m$.