منابع مشابه
On algebraically defined graphs in three dimensions
Let F be a field of characteristic zero or of a positive odd characteristic p. For a polynomial f ∈ F[x, y], we define a graph ΓF(xy, f) to be a bipartite graph with vertex partition P ∪L, P = F = L, and (p1, p2, p3) ∈ P is adjacent to [l1, l2, l3] ∈ L if and only if p2 + l2 = p1l1 and p3 + l3 = f(p1, l1). If f = xy, the graph ΓF(xy, xy ) has the length of a shortest cycle (the girth) equal to ...
متن کاملNote on generalizing theorems in algebraically closed fields
The generalization properties of algebraically closed elds ACF p of characteristic p > 0 and ACF 0 of characteristic 0 are investigated in the sequent calculus with blocks of quantiiers. It is shown that ACF p admits nite term bases, and ACF 0 admits term bases with primality constraints. From these results the analogs of Kreisel's Conjecture for these theories follow: If for some k, A(1 + + 1)...
متن کاملConnectivity of some Algebraically Defined Digraphs
Let p be a prime, e a positive integer, q = pe, and let Fq denote the finite field of q elements. Let fi : Fq → Fq be arbitrary functions, where 1 6 i 6 l, i and l are integers. The digraph D = D(q; f), where f = (f1, . . . , fl) : Fq → Fq, is defined as follows. The vertex set of D is Fl+1 q . There is an arc from a vertex x = (x1, . . . , xl+1) to a vertex y = (y1, . . . , yl+1) if xi + yi = ...
متن کاملOn the uniqueness of some girth eight algebraically defined graphs
Let F be a field. For a polynomial f ∈ F[x, y], we define a bipartite graph ΓF(f) with vertex partition P ∪ L, P = F = L, and (p1, p2, p3) ∈ P is adjacent to [l1, l2, l3] ∈ L if and only if p2 + l2 = p1l1 and p3 + l3 = f(p1, l1). It is known that the graph ΓF(xy ) has no cycles of length less than eight. The main result of this paper is that ΓF(xy ) is the only graph ΓF(f) with this property wh...
متن کاملSpectral and Combinatorial Properties of Some Algebraically Defined Graphs
Let k ≥ 3 be an integer, q be a prime power, and Fq denote the field of q elements. Let fi, gi ∈ Fq[X], 3 ≤ i ≤ k, such that gi(−X) = − gi(X). We define a graph S(k, q) = S(k, q; f3, g3, · · · , fk, gk) as a graph with the vertex set Fq and edges defined as follows: vertices a = (a1, a2, . . . , ak) and b = (b1, b2, . . . , bk) are adjacent if a1 6= b1 and the following k− 2 relations on their ...
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ژورنال
عنوان ژورنال: Ricerche di Matematica
سال: 2020
ISSN: 0035-5038,1827-3491
DOI: 10.1007/s11587-020-00535-3