Directed Graph Of Order Research Articles

Realization of digraphs in Abelian groups and its consequences

AbstractLet be a directed graph of order with no component of order less than 3, and let be a finite Abelian group such that or if is large enough with respect to an arbitrarily fixed then . We show that there exists an injective mapping from to the group such that for every connected component of , where 0 is the identity element of . Moreover we show some applications of this result to group distance magic labelings.

Odd Vertex Out-Magic Total Labeling of Some 2-Regular Digraphs

Let D(V, A) be a directed graph of order p and size q. An Odd vertex out-magic total labeling (OVOMTL) of a graph D is bijection f : V(D) ∪ A(D) → {1, 2, … p + q} in order f (V(D)) = {1, 3, …, 2p − 1}, for all υ ∈ V (D), , a constant. A uGo(v) digraph which admits an OVOMTL is termed as Odd Vertex out-magic total (OVOMT). Herein, we emphasize on feature of some 2-regular directed graphs that are Odd Vertex out-magic total (OVOMT).

Disjoint directed cycles in directed graphs

Let D=(V,A) be a directed graph of order n≥4. Let a(v) denote the degree of v in D for all v∈V. Suppose that a(x)+a(y)≥3n−4 for all {x,y}⊆V with x≠y. Then for any k integers n1,…,nk with ni≥2(1≤i≤k) and n1+⋯+nk≤n, D contains k disjoint directed cycles C1,…,Ck of orders n1,…,nk, respectively, or D belongs to one known class of directed graphs. This confirms the conjecture in Wang (2000) as a corollary.

Spanning Trees of Dense Directed Graphs

In the nineties, Komlós, Sárközy and Szemerédi confirmed a conjecture of Bollobás, showing that for every positive α, Δ and sufficiently large n, every graph with minimum degree (1/2 + α)n contains every tree of order n and maximum degree at most Δ. We obtain a directed graph analogue of their result, where the minimum degree is replaced by minimum semidegree (which is the minimum of all in- and out-degrees over all vertices) and the maximum degree is replaced by the maximum degree of the underlying graph (i.e., the maximum degree of the graph we obtain by ignoring the orientation of edges).In fact, we prove a stronger result, which states a sufficient condition for a tree of order n to be contained in every directed graph of order n and minimum semidegree (1/2 + α)n. This result implies that for all positive real α and sufficiently large n every directed graph of order n with minimum semidegree (1/2+α)n contains almost every spanning oriented tree T of order n.

ScEpath: energy landscape-based inference of transition probabilities and cellular trajectories from single-cell transcriptomic data.

MotivationSingle-cell RNA-sequencing (scRNA-seq) offers unprecedented resolution for studying cellular decision-making processes. Robust inference of cell state transition paths and probabilities is an important yet challenging step in the analysis of these data.ResultsHere we present scEpath, an algorithm that calculates energy landscapes and probabilistic directed graphs in order to reconstruct developmental trajectories. We quantify the energy landscape using ‘single-cell energy’ and distance-based measures, and find that the combination of these enables robust inference of the transition probabilities and lineage relationships between cell states. We also identify marker genes and gene expression patterns associated with cell state transitions. Our approach produces pseudotemporal orderings that are—in combination—more robust and accurate than current methods, and offers higher resolution dynamics of the cell state transitions, leading to new insight into key transition events during differentiation and development. Moreover, scEpath is robust to variation in the size of the input gene set, and is broadly unsupervised, requiring few parameters to be set by the user. Applications of scEpath led to the identification of a cell-cell communication network implicated in early human embryo development, and novel transcription factors important for myoblast differentiation. scEpath allows us to identify common and specific temporal dynamics and transcriptional factor programs along branched lineages, as well as the transition probabilities that control cell fates.Availability and implementationA MATLAB package of scEpath is available at https://github.com/sqjin/scEpath.Supplementary information Supplementary data are available at Bioinformatics online.

Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree greater than $${\frac{1}{2}n + 7\sqrt{n}/3}$$ contains an anti-directed hamiltonian cycle. In addition, we show that D contains anti-directed cycles of all possible (even) lengths when n is sufficiently large and has minimum in- and out-degree at least $${(1/2+ \epsilon)n}$$ for any $${\epsilon > 0}$$ .

Vertex-Oriented Hamilton Cycles in Directed Graphs

Let $D$ be a directed graph of order $n$. An anti-directed Hamilton cycle $H$ in $D$ is a Hamilton cycle in the graph underlying $D$ such that no pair of consecutive arcs in $H$ form a directed path in $D$. We prove that if $D$ is a directed graph with even order $n$ and if the indegree and the outdegree of each vertex of $D$ is at least ${2\over 3}n$ then $D$ contains an anti-directed Hamilton cycle. This improves a bound of Grant. Let $V(D) = P \cup Q$ be a partition of $V(D)$. A $(P,Q)$ vertex-oriented Hamilton cycle in $D$ is a Hamilton cycle $H$ in the graph underlying $D$ such that for each $v \in P$, consecutive arcs of $H$ incident on $v$ do not form a directed path in $D$, and, for each $v \in Q$, consecutive arcs of $H$ incident on $v$ form a directed path in $D$. We give sufficient conditions for the existence of a $(P,Q)$ vertex-oriented Hamilton cycle in $D$ for the cases when $|P| \geq {2\over 3}n$ and when ${1\over 3}n \leq |P| \leq {2\over 3}n$. This sharpens a bound given by Badheka et al.

Oriented Forests in Directed Graphs

Abstract If H is any oriented forest with m edges, then any directed graph of order ⩾ | H | with minimum out-degree as well as minimum in-degree ⩾m contains H.