Bidirected graphs are multigraphs where every edge has an independent direction at each end. In the paper, with an arbitrary bidirected graph we associate a non-negative integral quadratic form (called the incidence form of the graph), and determine all forms that appear in this way in two main results: first, among non-negative connected unit forms, precisely those of Dynkin type $\mathbb{A}$ or $\mathbb{D}$ are incidence forms; second, we give simple conditions on the coefficients of a non-negative connected non-unitary form to be an incidence form. We say that those non-unitary forms have Dynkin type $\mathbb{C}$, and justify such nomenclature by generalizing known classifications and properties of non-negative quadratic forms of Dynkin types $\mathbb{A}$ and $\mathbb{D}$ to the introduced type~$\mathbb{C}$. We also show that the graphical framework of an incidence form is an useful tool to visualize its arithmetical properties, to prove new facts and to perform efficient computations for integral quadratic forms and related problems in number theory, algebra and graph theory. For instance, in a third main result we relate the walks of a bidirected graph with the $0,1,2$-roots of the associated incidence form (and to the classical root systems in the positive case). Moreover, we prove the universality property for a large class of integral quadratic forms, provide computational methods to find solutions or to characterize the finiteness of the sets of solutions of various related Diophantine equations, show a variant of Whitney's theorem on line graphs using switching classes, and apply our techniques to give a conceptual and constructive proof of the non-negativity (and possible Dynkin types) of the Euler quadratic forms of a class of finite-dimensional gentle algebras.
