I am interested in upper bounding the second largest eigenvalue of the adjacency matrix of a graph $T$ with the following property: 1. $T$ contains self loops. 2. $T$ contains multiple edges (of, alternatively, is a weighted graph). 3. $T$ has a structure of a perfect binary tree (with self loops in every vertex).
The eigenvalue bounds that arise from edge expansion and vertex expansion are too weak for the bound I am trying to prove. How would you suggest dealing with weighted binary trees?