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The Rado graph, an infinite graph that models exactly the first-order sentences that are almost always true of finite graphs. Glebskiĭ et al. (1969) and, independently, Fagin (1976) proved a zero–one law for first-order graph logic; Fagin's proof used the compactness theorem.
A fundamental result in this area, proved independently by Glebskiĭ et al. and by Ronald Fagin, is that there is a zero-one law for (, /) for every property that can be described in the first-order logic of graphs. [2] Moreover, the limiting probability is one if and only if the infinite Rado graph has the
As elsewhere in graph theory, the order-zero graph (graph with no vertices) is generally not considered to be a tree: while it is vacuously connected as a graph (any two vertices can be connected by a path), it is not 0-connected (or even (−1)-connected) in algebraic topology, unlike non-empty trees, and violates the "one more vertex than ...
Graph order, the number of nodes in a graph; First order and second order logic of graphs; Topological ordering of directed acyclic graphs; Degeneracy ordering of undirected graphs; Elimination ordering of chordal graphs; Order, the complexity of a structure within a graph: see haven (graph theory) and bramble (graph theory)
first order The first order logic of graphs is a form of logic in which variables represent vertices of a graph, and there exists a binary predicate to test whether two vertices are adjacent. To be distinguished from second order logic, in which variables can also represent sets of vertices or edges.-flap
This is not a first-order axiomatization as one of Hilbert's axioms is a second order completeness axiom. Tarski's axioms are a first-order axiomatization of Euclidean geometry. Tarski showed this axiom system is complete and decidable by relating it to the complete and decidable theory of real closed fields.
The order of a graph is ... The first textbook on graph theory was written by ... 1969; Chinese, Shanghai 1963; Second printing of the 1962 first English edition ...
An edgeless graph is occasionally referred to as a null graph in contexts where the order-zero graph is not permitted. [1] [2] It is a 0-regular graph. The notation K n arises from the fact that the n-vertex edgeless graph is the complement of the complete graph K n.