Abstract:
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbits on the graph. This includes trace formulae with, respectively, absolutely and conditionally convergent periodic orbit sums; the convergence depending on properties of the test functions used. We also prove a trace formula for the heat kernel and provide small-$t$ asymptotics for the trace of the heat kernel.

Abstract:
It is a well-known fact that the first and last non-trivial coefficients of the characteristic polynomial of a linear operator are respectively its trace and its determinant. This work shows how to compute recursively all the coefficients as polynomial functions in the traces of successive powers of the operator. With the aid of Cayley-Hamilton's theorem the trace formulas provide a rational formula for the resolvent kernel and an operator-valued null identity for each finite dimension of the underlying vector space. The 4-dimensional resolvent formula allows an algebraic solution of the inverse metric problem in general relativity.

Abstract:
We give a new proof of the trace formula for regular graphs. Our approach is inspired by path integral approach in quantum mechanics, and calculations are mostly combinatorial.

Abstract:
The paper is a continuation of the study started in \cite{Yorzh1}. Schrodinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of $\delta$ type. Either an infinite series of trace formulae (provided that edge potentials are infinitely smooth) or a finite number of such formulae (in the cases of $L_1$ and $C^M$ edge potentials) are obtained which link together two different quantum graphs under the assumption that their spectra coincide. Applications are given to the problem of recovering matching conditions for a quantum graph based on its spectrum.

Abstract:
In this paper, we define the notion of graph trace kernels as a generalization of trace kernels. We associate a microlocal Lefschetz class with a graph trace kernel and prove that this class is functorial with respect to the composition of kernels. We apply graph trace kernels to the microlocal Lefschetz fixed point formula for constructible sheaves.

Abstract:
Several general results for the spectral determinant of the Schr\"odinger operator on metric graphs are reviewed. Then, a simple derivation for the $\zeta$-regularised spectral determinant is proposed, based on the Roth trace formula. Two types of boundary conditions are studied: functions continuous at the vertices and functions whose derivative is continuous at the vertices. The $\zeta$-regularised spectral determinant of the Schr\"odinger operator acting on functions with the most general boundary conditions is conjectured in conclusion. The relation to the Ihara, Bass and Bartholdi formulae obtained for combinatorial graphs is also discussed.

Abstract:
Differential operators on metric graphs are investigated. It is proven that vertex matching (boundary) conditions can be successfully parameterized by the vertex scattering matrix. Two new families of matching conditions are investigated: hyperplanar Neumann and hyperplanar Dirichlet conditions. Using trace formula it is shown that the spectrum of the Laplace operator determines certain geometric properties of the underlying graph.

Abstract:
The article surveys quantization schemes for metric graphs with spin. Typically quantum graphs are defined with the Laplace or Schrodinger operator which describe particles whose intrinsic angular momentum (spin) is zero. However, in many applications, for example modeling an electron (which has spin-1/2) on a network of thin wires, it is necessary to consider operators which allow spin-orbit interaction. The article presents a review of quantization schemes for graphs with three such Hamiltonian operators, the Dirac, Pauli and Rashba Hamiltonians. Comparing results for the trace formula, spectral statistics and spin-orbit localization on quantum graphs with spin Hamiltonians.

Abstract:
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on a metric graph, we define a new type of combinatorial Laplacian. With this generalised discrete Laplacian, it is possible to relate the spectral theory on discrete and metric graphs. Moreover, we describe a connection of metric graphs with manifolds. Finally, we comment on Cheeger's inequality and trace formulas for metric and discrete (generalised) Laplacians.

Abstract:
We study a simple stochastic differential equation driven by one Brownian motion on a general oriented metric graph whose solutions are stochastic flows of kernels. Under some condition, we describe the laws of all solutions. This work is a natural continuation of some previous papers by Hajri, Hajri-Raimond and Le Jan-Raimond where some particular graphs have been considered.