I try to find an upper bound for the mixing time of a random walk $S$ on a connected graph $L=(V,E)$ which has $k<\min_{v\in V}d(v)$ loops at every vertex. The transition probabilities of this random walk are given by $$p_{v,w}=\dfrac{1}{d(v)+k};\qquad p_{v,v}=\dfrac{k}{d(v)+k}$$ where $d(v)$ is the degree of the vertex $v$. I can easily calculate the stationary distribution $\pi(v) = \dfrac{d(v)+k}{\sum_{v}d(v)+k|V|}$ but now I am interested in the mixing time of $S$, $t_{mix}:=\inf\{t\geq 0|\inf_{\mu}||\mu P^t -\pi||_{TV}\leq 4^{-1}\}$ (see Markov Chains and Mixing Times; David A. Levin, Yuval Peres, Elizabeth L. Wilmer). For a random walk on a simple graph, without loops, I know that the mixing time is bounded from above by $C\log\left(\min_{v\in V}\dfrac{1}{\pi(v)}\right)\Phi(L)^{-1}$ where $C$ is some constant and $\Phi(L)$ is the cheeger constant of $L$. In all approaches I have found, which use spectral graph theory, the fact that $\mathrm{trace}(A)=0$ is explicitly used for the adjacency matrix $A$ of $L$. But how does it work for graphs with loops?

Thank you for your help!

(This question is a copy from here)