I discuss the work of many authors on various matrices used to study signed graphs, concentrating on adjacency and incidence matrices and the closely related topics of Kirchhoff (‘Laplacian’) matrices, line graphs, and very strong regularity.

In this paper‎, ‎we prove a multiplicity result for some biharmonic elliptic equation of Kirchhoff type and involving nonlinearities with critical exponential growth at infinity‎. ‎Using some variational arguments and exploiting the symmetries of the problem‎, ‎we establish a multiplicity result giving two nontrivial solutions‎.

The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., $\psi^{p-2}\psi$) on noncompact metric graphs a finite number of edges, in case Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for associated Cauchy problem operator domain and, infinite $N$-star graphs, existence standing waves bifurcating from trivial solution at $\omega=mc^2$, any ...

In this article, we study a Kirchhoff type problem with nonlinear Neumann boundary conditions on a bounded domain. By using variational methods, we prove the existence of infinitely many solutions.

A family of unified models in scattering from rough surfaces is based on local corrections of the tangent plane approximation through higher-order derivatives of the surface. We revisit these methods in a common framework when the correction is limited to the curvature, that is essentially the secondorder derivative. The resulting expression is formally identical to the weighted curvature appro...

We study a nonlinear p(x)-Kirchhoff type problem with Dirichlet boundary condition, in the case of reaction term depending also on gradient (convection). Using topological approach based Galerkin method, we discuss existence two notions solutions: strong generalized solution and weak solution. Strengthening bound Kirchhoff (positivity condition), establish solution, this time using theory opera...

Kirchhoff showed that the number of spanning trees a graph is spectral determinant combinatorial Laplacian divided by vertices; we reframe this result in quantum setting. We prove Laplace operator on finite connected metric with standard (Neumann–Kirchhoff) vertex conditions determines when lengths edges are sufficiently close together. To obtain result, analyze an equilateral whose spectrum cl...