In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given. Based on the cyclotomic numbers of finite fields, we present two infinite families of directed strongly regular Cayley graphs from semi-direct products of groups.
In this paper, we count acyclic and strongly connected uniform directed labeled hypergraphs. For these combinatorial structures, we introduce a specific generating function allowing us to recover and generalize some results on the number of directed acyclic graphs and the number of strongly connected directed graphs.
This is the second report of our work on the construction of directed strongly regular graphs. In our previous work, we constructed a couple of infinite families of new directed strongly regular graphs on the sets of antiflags of partial geometries and group divisible designs. In this paper, we use some collections of antiflags (not the entire set of antiflags) of tactical configurations to construct another couple of infinite families of directed strongly regular graphs. Our construction methods are capable of producing many, if not all, nonisomorphic directed strongly regular graphs with sam…
In this paper we introduce a construction of directed strongly regular graphs from smaller ones using equitable partitions. Each equitable partition of a single DSRG satisfying several conditions leads to an infinite family of directed strongly regular graphs. We construct in this way dozens of infinite families. For order at most 110, we confirm the existence of DSRGs for 30 previously open parameter sets.
Given an Eulerian digraph, we consider the genus distribution of its face-oriented embeddings. We prove that such distribution is log-concave for two families of Eulerian digraphs, thus giving a positive answer for these families to a question asked in Bonnington, Conder, Morton and McKenna (2002). Our proof uses real-rooted polynomials and the representation theory of the symmetric group $\mathbb{S}_n$. The result is also extended to some factorizations of the identity in $\mathbb{S}_n$ that are rotation systems of some families of one-face constellations.
Olmez, in "Symmetric $1\frac{1}{2}$-Designs and $1\frac{1}{2}$-Difference Sets" (2014), introduced the concept of a partial geometric difference set (also referred to as a $1\frac{1}{2}$-design), and showed that partial geometric difference sets give partial geometric designs. Nowak et al., in "Partial Geometric Difference Families" (2014), introduced the concept of a partial difference family, and showed that these also give partial geometric designs. It was shown by Brouwer et al. in "Directed strongly regular graphs from $1\frac{1}{2}$-designs" (2012) that directed strongly regular graphs c…
Etzion et al. introduced metrics on $\mathbb{F}_2^n$ based on directed graphs on $n$ vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between weighted poset metrics and directed graph based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices which admit th…
The family of Directed Acyclic Graphs as well as some related graphs are analyzed with respect to extremal behavior in relation with the family of intersection graphs for families of boxes with transverse intersection.
In an earlier work, the author together with Guo [Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory 85 (2017) 217-248] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it appears that a more natural Hermitian matrix exists, and it is the purpose of this note to bring this new Hermitian matrix to the attention of researchers in algebraic graph theory.
The paper is devoted to the study of lattice paths that consist of vertical steps $(0,-1)$ and non-vertical steps $(1,k)$ for some $k\in \mathbb Z$. Two special families of primary and free lattice paths with vertical steps are considered. It is shown that for any family of primary paths there are equinumerous families of proper weighted lattice paths that consist of only non-vertical steps. The relation between primary and free paths is established and some combinatorial and statistical properties are obtained. It is shown that the expected number of vertical steps in a primary path running f…
On the Graphical $r$-Stirling Numbers of the First Kind for Specific Graph Families
Department of Computer science, University of Torbat-e Jam（托尔巴特-贾姆大学计算机系）；Department of Pure Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad（马什哈德费尔多西大学理学院纯数学系）
We use finite incident structures to construct new infinite families of directed strongly regular graphs with parameters \[(l(q-1)q^l,\ l(q-1)q^{l-1},\ (lq-l+1)q^{l-2},\ (l-1)(q-1)q^{l-2},\ (lq-l+1)q^{l-2})\] for integers $q$ and $l$ ($q, l\ge 2$), and \[(lq^2(q-1),\ lq(q-1),\ lq-l+1,\ (l-1)(q-1),\ lq-l+1)\] for all prime powers $q$ and $l\in \{1, 2,..., q\}$. The new graphs given by these constructions have parameters $(36, 12, 5, 2, 5)$, $(54, 18, 7, 4, 7)$, $(72, 24, 10, 4, 10)$, $(96, 24, 7, 3, 7)$, $(108, 36, 14, 8, 14)$ and $(108, 36, 15, 6, 15)$ listed as feasible parameters on "Paramet…
A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow u_j$ exist for $1 \leq i < j \leq t$. In this paper, we study semi-transitivity of families of directed split graphs obtained by iterations of morphisms applied to the adjacency matrices and giving in the limit infinite directed split graphs. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. We fully classify sem…
We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpile group of LG by its k-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs.
In this paper, we study the Reconstruction Conjecture for finite simple graphs. Let $Γ$ and $Γ'$ be finite simple graphs with at least three vertices such that there exists a bijective map $f:V(Γ) \rightarrow V(Γ')$ and for any $v\in V(Γ)$, there exists an isomorphism $ϕ_v:Γ-v \to Γ'-f(v)$. Then we define the associated directed graph $\widetildeΓ=\widetildeΓ(Γ,Γ',f,\{ϕ_v\}_{v\in V(Γ)})$ with two kinds of arrows from the graphs $Γ$ and $Γ'$, the bijective map $f$ and the isomorphisms $\{ϕ_v\}_{v\in V(Γ)}$. By investigating the associated directed graph $\widetildeΓ$, we study when are the two …
In this paper, we study independent domination in directed graphs, which was recently introduced by Cary, Cary, and Prabhu. We provide a short, algorithmic proof that all directed acyclic graphs contain an independent dominating set. Using linear algebraic tools, we prove that any strongly connected graph with even period has at least two independent dominating sets, generalizing several of the results of Cary, Cary, and Prabhu. We prove that determining the period of the graph is not sufficient to determine the existence of an independent dominating set by constructing a few examples of infin…
A \textit{subtractive arc-magic labeling} (SAML) of a directed graph $G=(V,A)$ is a bijection $λ:V\cup A \to \{1,2,\ldots,|V|+|A|\}$ with the property that for every $xy\in A$ we have $λ(xy)+λ(y)-λ(x)$ equals to an integer constant. If $λ(xy)+λ(y)-λ(x)$ are distinct for every $xy\in A$, then $λ$ is a \textit{subtractive arc-antimagic labeling} (SAAL). A \textit{subtractive vertex-magic labeling} (SVML) of $G$ is such bijection with the property that for every $x\in V$ we have $λ(x)+\sum_{y\in V, yx \in A} λ(yx)-\sum_{y\in V, xy\in A} λ(xy)$ equals to an integer constant. If $λ(x)+\sum_{y\in V,…
Duval, in "A Directed Graph Version of Strongly Regular Graphs" [{\it Journal of Combinatorial Theory}, Series A 47 (1988) 71 - 100], introduced the concept of directed strongly regular graphs. In this paper we construct several rich families of directed strongly regular graphs with new parameters. Our constructions yielding new parameters are based on extending known explicit constructions to cover more parameter sets. We also explore some of the links between Cayley graphs, block matrices and directed strongly regular graphs with certain parameters. Directed strongly regular graphs which are…
The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts. To represent such signed directed graphs, we use a striking equivalence to $\mathbb{T}_6$-gain graphs to formulate a Hermitian adjacency matrix, whose entries are the unit Eisenstein integers $\exp(kπi/3),$ $k\in \mathbb{Z}_6.$ Many well-known results, such as (gain) switching and eigenvalue interlacing, naturally carry over to this paradigm. We show that n…
In this paper, various kinds of invariants of directed graphs are summarized. In the first topic, the invariant w(G) for a directed graph G is introduced, which is primarily defined by S. Chen and X.M. Chen to solve a problem of weak connectedness of tensor product of two directed graphs. Further, we present our recent studies on the invariant w(G) in categorical view. In the second topic, Homology theory on directed graph is introduced, and we also cast on categorical view of the definition. The third topic mainly focuses on Laplacians on graphs, including traditional work and latest result o…
We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, $q$-analogs (which we name $q$TNN and $q$SFL properties). %In our discussion, essential sets, introduced by Fult…
A $k$-ranking of a directed graph $G$ is a labeling of the vertex set of $G$ with $k$ positive integers such that every directed path connecting two vertices with the same label includes a vertex with a larger label in between. The rank number of $G$ is defined to be the smallest $k$ such that $G$ has a $k$-ranking. We find the largest possible directed graph that can be obtained from a directed path or a directed cycle by attaching new edges to the vertices such that the new graphs have the same rank number as the original graphs. The adjacency matrix of the resulting graph is embedded in the…
In a graph, a watchman's walk is a minimum closed dominating walk. Given a graph $G$ and a single watchman, the length of a watchman's walk in $G$ (the watchman number) is denoted by $w(G)$ and the typical goals of the watchman's walk problem is to determine $w(G)$ and find a watchman's walk in $G$. In this paper, we extend the watchman's walk problem to directed graphs. In a directed graph, we say that the watchman can only move to and see the vertices that are adjacent to him relative to outgoing arcs. That is, a watchman's walk is oriented and domination occurs in the direction of the arcs.…
We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new families of mixed graphs. For every fixed value of the degree, these families have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are show…
Fix an oriented graph H, and let G be a graph with bounded clique number and very large chromatic number. If we somehow orient its edges, must there be an induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for two specific kinds of digraph H: the three-edge path, with the first and last edges both directed towards the interior; and stars (with many edges directed out and many directed in). Aboulker et al subsequently conjectured that the answer is affirmative in both cases. We give affirmative answers to both questions.
We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality assumptions on the lengths of all closed orbits on the graph. In addition we assign transition probabilities to such graphs and compute statistics of the corresponding random walks. Some examples and applications are reviewed.
Erdös-Hajnal conjecture states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. This conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H-$free $n-$vertex tournament $T$ contains a transitive subtournament of order at least $ n^{ε(H)} $. This conjecture is known to hold for a few infinite families of tournaments. In this paper we construct two n…
In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, trees, DAGs, cycles, and bipartite graphs. We also provide the idomatic number for special cases of some of these families of digraphs.
For families of 4-regular directed circulant graphs with $n$ vertices, we count the number of primitive periodic orbits of length up to at least $n$. The relevant counting techniques are then extended to count the number of primitive pseudo orbits (sets of distinct primitive periodic orbits) of length up to at least $n$ that lack self-intersections, or that self-intersect only at individual vertices repeated exactly twice (2-encounters of length zero), for two particular families of 4-regular directed circulant graphs. We then regard these two families of graphs as families of quantum graphs a…
There is much recent interest in excluded subposets. Given a fixed poset $P$, how many subsets of $[n]$ can found without a copy of $P$ realized by the subset relation? The hardest and most intensely investigated problem of this kind is when $P$ is a diamond, i.e. the power set of a 2 element set. In this paper, we show infinitely many asymptotically tight constructions using random set families defined from posets based on Abelian groups. They are provided by the convergence of Markov chains on groups. Such constructions suggest that the diamond problem is hard.
In this paper, we enumerate two families of polycubes, the directed plateau polycubes and the plateau polycubes, with respect to the width and a new parameter, the Lateral Area. We give an explicit formula and the generating function for each of the two families of polycubes. Moreover, some asymptotic results about plateau polycubes are provided. We also establish results concerning the enumeration of column-convex polyominoes that are useful to get asymptotic results of polycubes.
In this paper, we defined three kinds of measures depending on the given finite directed graphs. For the given finite directed graph, we can construct the free semigroupoid, the diagram set and the reduced diagram set, as algebraic structures determined by the admissibility on the graph. The power sets of such structures are sigma algebras of them, respectively. On them, we can define a measure. The pain purpose of this paper is to introduce those three measures and observe some properties of them. Moreover, we consider Measure Theory on them
Alon and Yuster proved that the number of orientations of any $n$-vertex graph in which every $K_3$ is transitively oriented is at most $2^{\lfloor n^2/4\rfloor}$ for $n \geq 10^4$ and conjectured that the precise lower bound on $n$ should be $n \geq 8$. We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with $n$ vertices is the only $n$-vertex graph for which there are exactly $2^{\lfloor n^2/4\rfloor}$ such orientations.
Two years ago, we alarmed the scientific community about the large number of bad papers in the literature on {\it zero difference balanced functions}, where direct proofs of seemingly new results are presented in an unnecessarily lengthy and convoluted way. Indeed, these results had been proved long before and very easily in terms of difference families. In spite of our report, papers of the same kind continue to proliferate. Regrettably, a further attempt to put the topic in order seems unavoidable. While some authors now follow our recommendation of using the terminology of {\it partitioned …
We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially larger in $k$ than any previously constructed. In the directed case, these are within a linear factor in $k$ of the Moore bound.
We construct multiple $qt$-binomial coefficients and related multiple analogues of several celebrated families of special numbers in this paper. These multidimensional generalizations include the first and the second kind of $qt$-Stirling numbers, $qt$-Bell numbers, $qt$-Bernoulli numbers, $qt$-Catalan numbers and the $qt$--Fibonacci numbers. In the course of developing main properties of these extensions, we prove results that are significant in their own rights such as certain probability measures on the set of integer partitions.
Homomorphism duality pairs play crucial role in the theory of relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper we characterize the infinite-finite antichain dualities and infinite-finite dualities with trees or forest on the left hand side. This work builds on our earlier papers that gave several example…
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph ${\tt G}$ is the simplicial complex whose faces are the multipaths of ${\tt G}$. We compute the Euler characteristic, and associated generating function, of the multipath complex for some families of graphs, including transitive tournaments and complete bipartite graphs. Then, we compute the homotopy type of multipath complexes of linear graphs, polygons, small grids and transitive tournaments. We show that they are all contractible or wedges of spheres. We introduce a new technique for deco…
We give two algorithms computing representative families of linear and uniform matroids and demonstrate how to use representative families for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include - LONGEST DIRECTED CYCLE - MINIMUM EQUIVALENT GRAPH (MEG) - Algorithms on graphs of bounded treewidth -k-PATH, k-TREE, and more generally, k-SUBGRAPH ISOMORPHISM, where the k-vertex pattern graph is of constant treewidth.
We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let $D$ be a digraph and $f$ a labeling of its vertices with positive integers; denote by $S(v)$ the sum of labels over all neighbors of each vertex $v$. The labeling $f$ is called \emph{topological additive numbering} if $S(u) < S(v)$ for each arc $(u,v)$ of the digraph. The problem asks to find the minimum number $k$ for which $D$ has a topological additive numbering with labels belonging to $\{ 1, \ldots, k \}$, denoted by $η_t…
For fixed positive integers $n$ and $k$, the Kneser graph $KG_{n,k}$ has vertices labeled by $k$-element subsets of $\{1,2,\dots,n\}$ and edges between disjoint sets. Keeping $k$ fixed and allowing $n$ to grow, one obtains a family of nested graphs, each of which is acted on by a symmetric group in a way which is compatible with all of the other actions. In this paper, we provide a framework for studying families of this kind using the FI-module theory of Church, Ellenberg, and Farb, and show that this theory has a variety of asymptotic consequences for such families of graphs. These consequen…
