Intersections of Finitely Generated Subgroups of Free Groups and Resolutions de Graphs.
Let D = (V (D),A(D)) be a digraph and k ≥ 2 be an integer. A subset N of V (D) is k-independent if for every pair of vertices u, v ∈ N, we have d(u, v) ≥ k; it is l-absorbent if for every u ∈ V (D) − N, there exists v ∈ N such that d(u, v) ≤ l. A (k, l)-kernel of D is a k-independent and l-absorbent subset of V (D). A k-kernel is a (k, k − 1)-kernel. A digraph D is k-transitive if for any path x0x1 ・ ・ ・ xk of length k, x0 dominates xk. Hernández-Cruz [3-transitive digraphs, Discuss. Math. Graph...
L-zero-divisor graphs of L-commutative rings have been introduced and studied in [5]. Here we consider L-zero-divisor graphs of a finite direct product of L-commutative rings. Specifically, we look at the preservation, or lack thereof, of the diameter and girth of the L-ziro-divisor graph of a L-ring when extending to a finite direct product of L-commutative rings.
The girth of graphs on Weyl groups, with no restriction on the associated root system, is determined. It is shown that the girth, when it is defined, is 3 except for at most four graphs for which it does not exceed 4.