Extremal spectral gap of regular graphs with bounded vertex connectivity

Abstract: The well-known inequality of Fiedler ensures that the spectral gap $r-λ_2(G)$ of a connected non-complete $r$-regular graph $G$ is bounded from above by the vertex connectivity $κ(G)$ of $G$. We prove that, for integers $t\geq2$ and $r&gt;2t<sup>2$, every connected $r$-regular graph with vertex connectivity at most $2t$ has spectral gap at most $\frac{1}{2}(r+t+2-\sqrt{(r-t+2)<sup>2-4t(t-1)})$. We show that this bound is nearly optimal for each fixed $t \ge 2$ and sufficiently large $r &gt; 2t<sup>2$ such that $r+1$ is divisible by $t$. We further prove that if $r&gt;(2t-1)(2t-2)$, then every connected $r$-regular graph whose vertex connectivity is odd and no more than $2t-1$ has spectral gap at most $r-\min\bigl{ξ(r,t),ν(r,t)\bigr}$ for some explicitly given functions $ξ(r,t)$ and $ν(r,t)$. If $r\geq6t<sup>2$, then this minimum equals $ξ(r,t)=\frac{r(2r-4t+5)}{2(r-t+2)}$. In particular, when $κ(G)=2t-1$, this gives $r-λ_2(G)&lt;t=(κ(G)+1)/2$, which improves Fiedler's bound by nearly a factor of two. Our upper bounds in their parameter ranges improve two known bounds on the spectral gap of regular graphs.