Loop conditions with strongly connected graphs (original) (raw)
Preface to the Special Issue “Algebraic Structures and Graph Theory”
Mathematics
Connections between algebraic structure theory and graph theory have been established in order to solve open problems in one theory with the help of the tools existing in the other, emphasizing the remarkable properties of one theory with techniques involving the second [...]
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.
On k-strong and k-cyclic digraphs
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.
Some remarks on cycles in graphs and digraphs
Discrete Mathematics, 2001
We survey several recent results on cycles of graphs and directed graphs of the following form: 'Does there exist a set of cycles with a property P that generates all the cycles by operation O?'.
Consistent Cycles in Graphs and Digraphs
Graphs and Combinatorics, 2007
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.
On Strongly Multiplicative Graphs
2012
A graph G with p vertices and q edges is said to be strongly multiplicative if the vertices are assigned distinct numbers 1, 2, 3, …, p such that the labels induced on the edges by the product of the end vertices are distinct. We prove some of the special graphs obtained through graph operations such as C n + (a graph obtained by adding pendent edge for each vertex of the cycle C n), (P n mK 1) +N 2 , P n + mK 1 and C n d (cycle C n with non-intersecting chords) are strongly multiplicative.
Cartesian products of directed graphs with loops
Discrete Mathematics, 2018
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.
Realizable cycle structures in digraphs
European Journal of Combinatorics, 2023
Simple cycles on a digraph form a trace monoid under the rule that two such cycles commute if and only if they are vertex disjoint. This rule describes the spatial conguration of simple cycles on the digraph. Cartier and Foata have showed that all combinatorial properties of closed walks are dictated by this trace monoid. We nd that most graph properties can be lost while maintaining the monoidal structure of simple cycles and thus cannot be inferred from it, including vertex-transitivity, regularity, planarity, Hamiltonicity, graph spectra, degree distribution and more. Conversely we nd that even allowing for multidigraphs, many congurations of simple cycles are not possible at all. The problem of determining whether a given conguration of simple cycles is realizable is highly non-trivial. We show at least that it is decidable and equivalent to the existence of integer solutions to systems of polynomial equations.
A Note on the Isomorphism Problem for Monomial Digraphs
Journal of Interconnection Networks
Let p be a prime e be a positive integer, [Formula: see text], and let [Formula: see text] denote the finite field of q elements. Let [Formula: see text], [Formula: see text], be integers. The monomial digraph [Formula: see text] is defined as follows: the vertex set of D is [Formula: see text], and [Formula: see text] is an arc in D if [Formula: see text]. In this note we study the question of isomorphism of monomial digraphs [Formula: see text] and [Formula: see text]. Several necessary conditions and several sufficient conditions for the isomorphism are found. We conjecture that one simple sufficient condition is also a necessary one.
Algebraic properties of line digraphs.
Let G be a digraph, L(G) its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. In addition, we study the spectra of those matrices and obtain a relationship between their characteristic polynomials that allows us to relate properties of G and L(G), specifically the number of cycles of a given length.
A characterization of strongly chordal graphs
Discrete Mathematics, 1998
In this paper, we present a simple charactrization of strongly chordal graphs. A chordal graph is strongly chordal if and only if every cycle on six or more vertices has an induced triangle with exactly two edges of the triangle as the chords of the cycle. (~) 1998 Elsevier Science B.V. All rights reserved
Graph cycles and diagram commutativity
Bases are exhibited for K n , K p,q , and Q d , and it is shown how each cycle of the various graphs can be built as a hierarchical ordered sum in which all of the partial sums are (simple) cycles with each cycle from either the basis or one of the hierarchically constructed cycle-sets meeting the partial sum of its predecessors in a nontrivial path. A property that holds for this " connected sum " of two cycles whenever it holds for both the parents is called constructable. It is shown that any constructable property holding for the specified basis cycles holds for every cycle in the graph, that commutativity is a constructable property of cycles in a groupoid diagram, and that " economies of scale " apply to ensuring commutativity for diagrams of the above three types. A procedure is given to extend a commutative groupoid diagram for any digraph that contains the diagram's scheme.
Some cycle and path related strongly*-graphs
International Journal of Mathematics Trends and Technology, 2017
Abstact -A graph with n vertices is said to be strongly * -graph if its vertices can be assigned the values {1, 2, . . . , n} in such a way that when an edge whose end vertices are labeled i and j, is labeled with the value i + j + ij such that all edges have distinct labels. Here we derive different strongly * -graphs in context of some graph operations.
The Cartesian product of graphs with loops
Ars Mathematica Contemporanea, 2015
We extend the definition of the Cartesian product to graphs with loops and show that the Sabidussi-Vizing unique factorization theorem for connected finite simple graphs still holds in this context for all connected finite graphs with at least one unlooped vertex. We also prove that this factorization can be computed in O(m) time, where m is the number of edges of the given graph.
arXiv (Cornell University), 2017
The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by the graph complement operation it is isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the integers by demonstrating that the strong ring is homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth formula holds on the strong ring so that the Poincaré polynomial is compatible with the ring structure. The Zykov ring has the clique number as a ring homomorphism. Furthermore, the Cartesian ring has the property that the functor which attaches to a graph the spectrum of its connection Laplacian is multiplicative. The reason is that the connection Laplacians do tensor under multiplication, similarly to what the adjacency matrix does for the weak ring. The strong ring product of two graphs contains both the weak and direct product graphs as subgraphs. The Zykov, Sabidussi or Stanley-Reisner rings are so manifestations of a network arithmetic which has remarkable cohomological properties, dimension and spectral compatibility but where arithmetic questions like the complexity of detecting primes or factoring are not yet studied well. We illustrate the Zykov arithmetic with examples, especially from the subring generated by point graphs which contains spheres, stars or complete bipartite graphs. While things are formulated in the language of graph theory, all constructions generalize to the larger category of finite abstract simplicial complexes.
Structural and spectral properties of minimal strong digraphs
Electronic Notes in Discrete Mathematics
In this article, we focus on structural and spectral properties of minimal strong digraphs (MSDs). We carry out a comparative study of properties of MSDs versus trees. This analysis includes two new properties. The first one gives bounds on the coefficients of characteristic polynomials of trees (double directed trees), and conjectures the generalization of these bounds to MSDs. As a particular case, we prove that the independent coemcient of the characteristic polynomial of a tree or an MSD must be-1, 0 or 1. For trees, this fact means that a tree has at most one perfect matching; for MSDs, it means that an MSD has at most one covering by disjoint cycles. The property states that every MSD can be decomposed in a rooted spanning tree and a forest of reversed rooted trees, as factors. In our opinión, the analogies described suppose a significative change in the traditional point of view about this class of digraphs.