Extremal Digraphs Research Articles

On skew Laplacian spectrum and energy of digraphs

We consider the skew Laplacian matrix of a digraph [Formula: see text] obtained by giving an arbitrary direction to the edges of a graph [Formula: see text] having [Formula: see text] vertices and [Formula: see text] edges. With [Formula: see text] to be the skew Laplacian eigenvalues of [Formula: see text], the skew Laplacian energy [Formula: see text] of [Formula: see text] is defined as [Formula: see text]. In this paper, we analyze the effect of changing the orientation of an induced subdigraph on the skew Laplacian spectrum. We obtain bounds for the skew Laplacian energy [Formula: see text] in terms of various parameters associated with the digraph [Formula: see text] and the underlying graph [Formula: see text] and we characterize the extremal digraphs attaining these bounds. We also show these bounds improve some known bounds for some families of digraphs. Further, we show the existence of some families of skew Laplacian equienergetic digraphs.

Extremal digraphs avoiding an orientation of [formula omitted

Let P2,2 be the orientation of C4 which consists of two 2-paths with the same initial and terminal vertices. In this paper, we determine the maximum size of P2,2-free digraphs of order n as well as the extremal digraphs attaining the maximum size when n≥13.

Extremal digraphs avoiding distinct walks of length 4 with the same endpoints

Extremal digraphs avoiding distinct walks of length 4 with the same endpoints

Extremal digraphs for an upper bound on the Roman domination number

Let $$D=(V,A)$$ be a digraph. A Roman dominating function of a digraph D is a function f : $$V\longrightarrow \{0,1,2\}$$ such that every vertex u for which $$f(u)=0$$ has an in-neighbor v for which $$f(v)=2$$ . The weight of a Roman dominating function is the value $$f(V)=\sum _{u\in V}f(u)$$ . The minimum weight of a Roman dominating function of a digraph D is called the Roman domination number of D, denoted by $$\gamma _{R}(D)$$ . In this paper, we characterize some special classes of oriented graphs, namely out-regular oriented graphs and tournaments satisfying $$\gamma _{R}(D)=n-\Delta ^{+}(D)+1$$ . Moreover, we characterize digraphs D for which the equality $$\gamma _{R}(D)+\gamma _{R}(\overline{D})=n+3$$ holds, where $$\overline{D}$$ is the complement of D. Finally, we prove that the problem of deciding whether an oriented graph D satisfies $$\gamma _{R}(D)=n-\Delta ^{+}(D)+1$$ is CO- $$\mathcal {NP}$$ -complete.

A Turán problem on digraphs avoiding distinct walks of a given length with the same endpoints

Let k≥4 and n≥k+4 be positive integers. We determine the maximum size of digraphs of order n that avoid distinct walks of length k with the same endpoints. We also characterize the extremal digraphs attaining this maximum number when k≥5.

Extremal Digraphs for an Upper Bound on the Double Roman Domination Number

A doubleRoman dominating function of a digraph D is a function $$f:V\longrightarrow \{0,1,2,3\}$$ such that every vertex u for which $$f(u)=0$$ has an in-neighbor v for which $$f(v)=3$$ or at least two in-neighbors assigned 2 under f, while if $$f(u)=1$$, then the vertex u must have at least one in-neighbor assigned 2 or 3. The weight of a double Roman dominating function is the value $$f(V)=\sum _{u\in V}f(u)$$. The minimum weight of a double Roman dominating function of a digraph D is called the double Roman domination number of D, denoted by $$\gamma _{dR}\left( D\right) $$. This paper gives a descriptive characterization for some classes of digraphs satisfying $$\gamma _{dR}(D)=2\left( n-\Delta ^{+}(D)\right) +1$$. Also, a descriptive characterization for digraphs D of order $$n\ge 4$$ for which $$\gamma _{dR}(D)+\gamma _{dR}({\overline{D}})=2n+3$$ holds where $${\overline{D}}$$ is the complement of D.

On the lower bound of [formula omitted]-maximal digraphs

For a digraph D, let λ(D) be the arc-strong-connectivity of D. For an integer k>0, a simple digraph D with |V(D)|≥k+1 is k-maximal if every subdigraph H of D satisfies λ(H)≤k but for adding new arc to D results in a subdigraph H′ with λ(H′)≥k+1. We prove that if D is a simple k-maximal digraph on n>k+1≥2 vertices, then |A(D)|≥n2+(n−1)k+⌊nk+2⌋(1+2k−k+22). This bound is best possible. Furthermore, all extremal digraphs reaching this lower bound are characterized.

Maximum size of digraphs with some parameters

In this paper, we determine the maximum sizes of strong digraphs under the constraint that their some parameters are fixed, such as vertex connectivity, edge-connectivity, the number of cut vertices. The corresponding extremal digraphs are also characterized. In addition, we establish Nordhaus---Gaddum type theorem for the diameter when $$\overrightarrow{K_n}$$Knź decomposing into many parts. We also pose a related conjecture for Wiener index of digraphs.

The maximum Perron roots of digraphs with some given parameters

We characterize the extremal digraphs which attain the maximum Perron root of digraphs with given arc connectivity and number of vertices. We also characterize the extremal digraphs which attain the maximum Perron root of digraphs given diameter and number of vertices.

Sugary Foods and the Risk of Upper Respiratory Infections

We characterize the extremal digraphs which attain the maximum Perron root of digraphs with given arc connectivity and number of vertices. We also characterize the extremal digraphs which attain the maximum Perron root of digraphs given diameter and number of vertices.

Extremal digraphs with given clique number

We classify those digraphs with a given number of vertices and a given clique number that maximize the Perron root of the adjacency matrix.

CLASSIFICATION OF TWO-REGULAR DIGRAPHS WITH MAXIMUM DIAMETER

The Klee-Quaife problem is finding the minimum order <TEX>${\mu}(d,c,v)$</TEX> of the <TEX>$(d,c,v)$</TEX> graph, which is a <TEX>$c$</TEX>-vertex connected <TEX>$v$</TEX>-regular graph with diameter <TEX>$d$</TEX>. Many authors contributed finding <TEX>${\mu}(d,c,v)$</TEX> and they also enumerated and classied the graphs in several cases. This problem is naturally extended to the case of digraphs. So we are interested in the extended Klee-Quaife problem. In this paper, we deal with an equivalent problem, finding the maximum diameter of digraphs with given order, focused on 2-regular case. We show that the maximum diameter of strongly connected 2-regular digraphs with order <TEX>$n$</TEX> is <TEX>$n-3$</TEX>, and classify the digraphs which have diameter <TEX>$n-3$</TEX>. All 15 nonisomorphic extremal digraphs are listed.

Some results on the spectral radius of generalized [formula omitted] and [formula omitted]-digraphs

Let ∞∼-digraph be a generalized strongly connected ∞-digraph and let θ∼1-digraph and θ∼2-digraph be two kinds of generalized strongly connected θ-digraphs. In this paper, we characterize the extremal digraphs which achieve the maximum and minimum spectral radius among all Administrator in ∞∼-digraphs and θ∼1-digraphs. At the same time, we also determine the extremal digraph which achieves the maximum spectral radius among all Administrator in θ∼2-digraphs. Furthermore, we show that any ∞∼(a,b,c)-digraph is determined by the spectrum.

Extremal digraphs whose walks with the same initial and terminal vertices have distinct lengths

Let D be a digraph of order n in which any two walks with the same initial vertex and the same terminal vertex have distinct lengths. We prove that D has at most (n+1)2/4 arcs if n is odd and n(n+2)/4 arcs if n is even. The digraphs attaining this maximum size are determined.

Distance spectral radius of digraphs with given connectivity

Let D(G⃗) denote the distance matrix of a strongly connected digraph G⃗. The largest eigenvalue of D(G⃗) is called the distance spectral radius of a digraph G⃗, denoted by ϱ(G⃗). Recently, many studies proposed the use of ϱ(G⃗) as a molecular structure description of alkanes. In this paper, we characterize the extremal digraphs with minimum distance spectral radius among all digraphs with given vertex connectivity and the extremal graphs with minimum distance spectral radius among all graphs with given edge connectivity. Moreover, we give the exact value of the distance spectral radius of those extremal digraphs and graphs. We also characterize the graphs with the maximum distance spectral radius among all graphs of fixed order with given vertex connectivity 1 and 2.

Characterizing Extremal Digraphs for Identifying Codes and Extremal Cases of Bondy’s Theorem on Induced Subsets

An identifying code of a (di)graph G is a dominating subset C of the vertices of G such that all distinct vertices of G have distinct (in)neighbourhoods within C. In this paper, we classify all finite digraphs which only admit their whole vertex set as an identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well-known theorem of Bondy on set systems, we classify the extremal cases for this theorem.

A note on the spectral characterization of strongly connected bicyclic digraphs

Let D be a digraph with vertex set V(D) and A be the adjacency matrix of D. In this paper, we characterize the extremal digraphs which achieve the maximum and minimum spectral radius among strongly connected bicyclic digraphs. Furthermore, we show that any strongly connected bicyclic digraph is determined by the spectrum.

EXTREMAL CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

Cayley digraphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m*(d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k-element subset A of Γ such that diam ( Cay (Γ, A)) ≤ d, where diam ( Cay (Γ, A)) denotes the diameter of the Cayley digraph Cay (Γ, A) of Γ generated by A. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam (ℤm, A)) ≤ d. In this paper, we prove, among other results, that [Formula: see text] for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley digraph is optimal with respect to its diameter and degree can be a cyclic group.

On the exponents of two-colored digraphs with two cycles†

We consider the special primitive two-colored digraphs whose uncolored digraph has n+s vertices and consists of one n-cycle and one n−t-cycle. We give some primitivity conditions and an upper bound on the exponent. Further, for the case s=0, we give a tight upper bound on the exponent and the characterization of extremal two-colored digraphs. †Research supported by NNSF of China (No. 10571163) and NSF of Shanxi (No. 20041010).

Local bases of primitive non-powerful signed digraphs

In 1994, Z. Li, F. Hall and C. Eschenbach extended the concept of the index of convergence from nonnegative matrices to powerful sign pattern matrices. Recently, Jiayu Shao and Lihua You studied the bases of non-powerful irreducible sign pattern matrices. In this paper, the local bases, which are generalizations of the base, of primitive non-powerful signed digraphs are introduced, and sharp bounds for local bases of primitive non-powerful signed digraphs are obtained. Furthermore, extremal digraphs are described.