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

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: $A(g,c,n)=\min\{δ(G)\,|\; G \in \mathcal{G}(g,c,n) \}$, $B(g,c,n)=\max\{δ(G)\,|\; G \in \mathcal{G}(g,c,n) \}$, $α(g,c,m)=\min\{δ(G)\,|\; \in \mathcal{H}(g,c,m) \}$ and $β(g,c,m)=\max\{δ(G)\,|\; G \in \mathcal{H}(g,c,m) \}$. In particular, we obtain bounds for $A(g,c,n)$ and $α(g,c,m)$, and we compute the precise value of $B(g,c,n)$ and $β(g,c,m)$ for all values of $g$, $c$, $n$ and $m$.

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