The first eigenvalue of the $p-$Laplacian on quantum graphs

We study the first eigenvalue of the $p-$Laplacian (with $1<p<\mathrm{\infty }$) on a quantum graph with Dirichlet or Kirchoff boundary conditions on the nodes. We find lower and upper bounds for this eigenvalue when we prescribe the total sum of the lengths of the edges and the number of Dirichlet nodes of the graph. Also we find a formula for the shape derivative of the first eigenvalue (assuming that it is simple) when we perturb the graph by changing the length of an edge. Finally, we study in detail the limit cases $p\to \mathrm{\infty }$ and $p\to 1$.