Loop conditions with strongly connected graphs (original) (raw)
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Algebraic characterization of strongly connected graphs
International Journal of Pure and Applied Mathematics, 2012
We introduce the notion of uniform rank. A matrix C has an uniform rank r if its (normal) rank(C) equals r and each of its minors of size r is non-zero. Then we prove that a directed graph (digraph) with n vertices is strongly connected if and only if its laplacian has an uniform rank n − 1. Finally we show how to test if a digraph is strongly connected with a single Gaussian Elimination on the digraph's laplacian.
Algebra universalis, 2020
We prove that an idempotent operation generates a loop from a strongly connected digraph containing directed closed walks of all lengths under very mild (local) algebraic assumptions. Using the result, we reprove the existence of weakest non-trivial idempotent equations, and that a finite strongly connected digraph of algebraic length 1 compatible with a Taylor operation has a loop.
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Discrete Mathematics, 1996
proved that there is no degree of strong connectivity which guarantees a cycle through two given vertices in a digraph. In this paper we consider a large family of digraphs, including symmetric digraphs (i.e. digraphs obtained from undirected graphs by replacing each edge by a directed cycle of length two), semicomplete bipartite digraphs, locally semicomplete digraphs and all digraphs that can be obtained from acyclic digraphs and those mentioned above, by repeated substitutions of digraphs from one of these classes for vertices. We prove that for every natural number k, every k-strong digraph D from the family above is k-cyclic, i.e. for every set X of k vertices of D, there exists a cycle of D containing all the vertices of X. In particular, this implies that every k-strong quasi-transitive digraph is k-cyclic.
Connectivity of some Algebraically Defined Digraphs
The Electronic Journal of Combinatorics
Let ppp be a prime, eee a positive integer, q=peq = p^eq=pe, and let mathbbFq\mathbb{F}_qmathbbFq denote the finite field of qqq elements. Let ficolonmathbbFq2tomathbbFqf_i\colon\mathbb{F}_q^2\to\mathbb{F}_qficolonmathbbFq2tomathbbFq be arbitrary functions, where 1leilel1\le i\le l1leilel, iii and lll are integers. The digraph D=D(q;bff)D = D(q;\bf{f})D=D(q;bff), where bff=f1,dotso,fl)colonmathbbFq2tomathbbFql{\bf f}=f_1,\dotso,f_l)\colon\mathbb{F}_q^2\to\mathbb{F}_q^lbff=f1,dotso,fl)colonmathbbFq2tomathbbFql, is defined as follows. The vertex set of DDD is mathbbFql+1\mathbb{F}_q^{l+1}mathbbFql+1. There is an arc from a vertex bfx=(x1,dotso,xl+1){\bf x} = (x_1,\dotso,x_{l+1})bfx=(x1,dotso,xl+1) to a vertex bfy=(y1,dotso,yl+1){\bf y} = (y_1,\dotso,y_{l+1})bfy=(y1,dotso,yl+1) if xi+yi=fi−1(x1,y1)x_i + y_i = f_{i-1}(x_1,y_1)xi+yi=fi−1(x_1,y_1) for all iii, 2leilel+12\le i \le l+12leilel+1. In this paper we study the strong connectivity of DDD and completely describe its strong components. The digraphs DDD are directed analogues of some algebraically defined graphs, which have been studied extensively and have many applications.
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Let Γ be a finite digraph and let G be a subgroup of the automorphism group of Γ. A directed cycle C of Γ is called G-consistent whenever there is an element of G whose restriction to C is the 1-step rotation of C. Consistent cycles in finite arc-transitive graphs were introduced by Conway in one of his public lectures. He observed that the number of G-orbits of G-consistent cycles of an arc-transitive group G is precisely one less than the valency of the graph. In this paper, we give a detailed proof of this result in a more general settings of arbitrary groups of automorphisms of graphs and digraphs.
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Cartesian products of directed graphs with loops
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We show that every nontrivial finite or infinite connected directed graph with loops and at least one vertex without a loop is uniquely representable as a Cartesian or weak Cartesian product of prime graphs. For finite graphs the factorization can be computed in linear time and space.