I am interested in the following problem, which came up while working on this paper about estimating Betti numbers of Kähler manifolds, where we were not able to solve it and had to resort to ugliness. The setup is the following:
Say you have an (undirected) weighted graph $G$ with nonnegative (strictly positive if you prefer) weights on the edges, and consider the "Laplacian"-like operator $L$ on functions $u:G\to \mathbb{R_{\ge 0}}$ where $L(u)(p)=\sum \mu(e_i) u(p_i)$, where $e_i$ ranges over the edges incident to $p$ and $p_i$ is the vertex on the other side, i.e. the function that assigns to a vertex the weighted sum of the values at its neighbors. (If the weights at each point add to 1, this is simply the Laplacian, perhaps up to sign and adding a multiple of the identity)
Question 1: What can be said about the largest Eigenvalue of $L$?
Clearly the constant function is not an Eigenfunction any more, and it is not obvious how much can be expected in this generality, apart from some maximum principle stuff.
In practice, I am interested only in a very special class of weighted graphs where the answer is easily conjectured:
Let $i<n$ and consider the graph $G$ whose vertices are $i$-element subsets $I \subset \{1, \ldots, n\}$, with an edge $I\to I'$ whenever $I$ and $I'$ differ by exactly one element. For $\lambda = (\lambda_1 , \ldots, \lambda_n)\ge 0$, we put a weighing on this graph by assigning weight $\lambda_i \lambda_j$ to edges that exchange $i$ and $j$ between their endpoints. Call this weighted graph $G_\lambda$.
Question 2: Normalizing say to $\sum \lambda_i^2=1$, for which $\lambda$ is the largest Eigenvalue of $L$ on $G_\lambda$ maximal?
Conjecture: The global optimizer (for any i,n) should be $\lambda_i\equiv const$ with eigenfunction $u\equiv const$.
This problem seems to have so much symmetry as to make it seem unreasonable that there is no simple argument. However, we didn't manage to see one. Writing it out, one obtains a fourth degree optimization problem which is separately symmetric quadratic in each of $\lambda$ and $u$, but I know of no good theory for these (perhaps due to my unfamiliarity with the literature).
Supporting evidence: The claimed bound provably holds in both edge cases $i=1$ and $i=\frac{n}{2}$, $n$ even. (exchanging $i$ with $n-i$ gives the same graph)
