I am looking at a finite connected graph and I would like to know what is the best [i.e. largest] constant $\lambda_p$ in $$ \sum_x f(x) =0 \implies \| \Delta^{-1} f\|_{\ell^p} \leq \lambda_p^{-1} \|f\|_{\ell^p} $$ If $p=2$, $\lambda_2$ would be the first non-zero eigenvalue of $\Delta$.
There is an obvious bound (see below) but I am certain the proper keywords would open a world of results. Any expression of $\lambda_p$ above in terms of $\lambda_2$ or the conductance/isoperimetric/Cheeger constant would be very welcome.
Just to fix notations, let $P$ be the simple random walk operator (i.e. $(Pf)(x) = deg(x)^{-1} \sum_{y \sim x} f(y)$ where $deg(x)$ is the degree of $x$ and $\sim$ denotes the neighbour relation) on the graph and (for me) $\Delta = I-P$.
The easy bound follows by observing that (for functions $f$ with $\sum_s f(x)=0$) $\Delta^{-1} = \sum_{n \geq 0} P^n$. Then, one has (again restricting to the space of functions with zero mean) $$ \|P\|_{\ell^2\to \ell^2} = 1- \lambda_2 <1 \text{ while } \|P\|_{\ell^1\to \ell^1} \leq 1 \text{ and } \|P\|_{\ell^\infty \to \ell^\infty} \leq 1 $$ So that, by Riesz-Thorin interpolation, $\|P\|_{\ell^p \to \ell^p} <1$ for any $p \in (1, \infty)$. This suffices to see that the above serie converges. More precisely, this gives (with the notations $\bar{p} = \max( p,p')$) and $\tfrac{1}{p'} = 1- \tfrac{1}{p}$) $$ \frac{1}{\lambda_p} \leq \frac{1}{1-(1-\lambda_2)^{2/\bar{p}}} \leq \frac{\bar{p}}{2\lambda_2} $$
