of pairs $(x, y)$ where $Q(x, y) > 0$ and $x \ne y$. Write $$c_2(x, y) := \pi(x) Q^2(x, y)$$ and note that for $(x, y) \in E$, we have $$c_2(x, y) 
\ge \pi(x) [Q(x, x) Q(x, y) + Q(x, y) Q(y, y)] \ge 2 a c
\,.$$ Define the inner product $
\ip{f_1, f_2} := \sum_x f_1(x) f_2(x) \pi(x)
$ on $\ell^2(V, \pi)$.

The case where $\pi$ is infinite is simpler, so we treat that first. Let $f$ be a function on $V$ with finite support. Let $x_0$ be a vertex where $|f|$ achieves its maximum. Then $$\|f\|_\infty = |f(x_0)|
\le
{1\over2} \sum_{(x, y) \in E} |f(x) - f(y)|
\le
{1\over2} \sum_{x, y \in V} c_2(x, y) |f(x) - f(y)| /(2a c)
\,,
\label e.1$$ where the factor $1/2$ arises from counting each pair $(x, y)$ in each order. Applying \[e\].1/ to the function $f^2$, we obtain (2 a c)\^2 f\_\^4 &([12]{} \_[x, y V]{} c\_2(x, y) |f(x) - f(y)| |f(x) + f(y)| )\^2 &([12]{} \_[x, y V]{} c\_2(x, y) \[f(x) - f(y)\]\^2) ([12]{} \_[x, y V]{} c\_2(x, y)\[f(x) + f(y)\]\^2) &= by the Cauchy-Schwarz inequality and some straightforward algebra. Therefore, if $\ip{f, f} \le 1$, we have $
