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Let F m×n denote the set of m×n matrices with entries in F. Then F m×n is a vector space over F. Vector addition is just matrix addition and scalar multiplication is defined in the obvious way (by multiplying each entry by the same scalar). The zero vector is just the zero matrix. The dimension of F m×n is mn. One possible choice of basis ...
A Paley graph of order is -regular with all other eigenvalues being , making Paley graphs an infinite family of Ramanujan graphs. More generally, let f ( x ) {\displaystyle f(x)} be a degree 2 or 3 polynomial over F q {\displaystyle \mathbb {F} _{q}} .
The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero polynomial, f(x) = 0, is the x-axis. In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero ...
A ray, in an infinite graph, is an infinite simple path with exactly one endpoint. The ends of a graph are equivalence classes of rays. reachability The ability to get from one vertex to another within a graph. reachable Has an affirmative reachability. A vertex y is said to be reachable from a vertex x if there exists a path from x to y ...
When M is the cycle matroid M(G) of a graph G, the characteristic polynomial is a slight transformation of the chromatic polynomial, which is given by χ G (λ) = λ c p M(G) (λ), where c is the number of connected components of G. When M is the bond matroid M*(G) of a graph G, the characteristic polynomial equals the flow polynomial of G.
Most results of the finite-dimensional case also hold in the infinite-dimensional case too, with some modifications. Differentiation can also be defined to functions of several variables (for example, t ∈ R n {\displaystyle t\in R^{n}} or even t ∈ Y , {\displaystyle t\in Y,} where Y {\displaystyle Y} is an infinite-dimensional vector space).
The order polynomial is a polynomial studied in mathematics, in particular in algebraic graph theory and algebraic combinatorics. The order polynomial counts the number of order-preserving maps from a poset to a chain of length n {\displaystyle n} .
A family F of graphs is said to be closed under the operation of taking minors if every minor of a graph in F also belongs to F. If F is a minor-closed family, then let S be the class of graphs that are not in F (the complement of F). According to the Robertson–Seymour theorem, there exists a finite set H of minimal elements in S.