  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.
