On the Dα spectral radius of strongly connected digraphs

Let G be a strongly connected digraph with distance matrix D(G) and let Tr(G) be the diagonal matrix with vertex transmissions of G. For any real ? ? [0, 1], define the matrix D?(G) as D?(G) = ?Tr(G) + (1-?)D(G). The D? spectral radius of G is the spectral radius of D?(G). In this paper, we first give some upper and lower bounds for the D? spectral radius of G and characterize the extremal digraphs. Moreover, for digraphs that are not transmission regular, we give a lower bound on the difference between the maximum vertex transmission and the D? spectral radius. Finally, we obtain the D? eigenvalues of the join of certain regular digraphs.

On Multitracking of First-Order MASs with Adaptive Coupling Strength

The problem of cluster consensus with multiple leaders is called multitracking. In this article, a sort of multitracking of first-order multiagent systems with adaptive coupling strength is studied by the application of adaptive strategy, and the delayed relation between various leaders and clusters is considered. To reach the clustered multitracking goal, a novel pinning-like control protocol with adaptive approach is designed according to the properties of network topology. In addition, the structure of the networked system is a weakly connected digraph. Some conditions are derived to ensure that the nodes in the same cluster reach the consensus via tracking their leader, while leaders will keep a delayed relation with the settled leader node as time goes on to form the required delay consensus.

A new sufficient condition for the existence of 3-kernels

Let D be a digraph and k be a positive integer. We say a subset N of V(D) is a k-kernel of D if it is both k-independent and (k − 1)-absorbent. A short chord of a closed trail C = (v0, v1, . . . , vt) is an arc a = (vi, vj) which does not belong to C and the distance from vi to vj in C is exactly two. The spacing between two chords e = (u, v) and f = (x, y) in C is the distance from u to x in C. A set of chords in a closed trail C has an odd spacing if at least two chords have an odd spacing. In this work, we prove that if D is a strongly connected digraph where every odd cycle has a short chord and every even closed trail has three short chords with an odd spacing, then D has a 3-kernel.

On the Manoussakis Conjecture for a Digraph to be Hamiltonian

Manoussakis (J. Graph Theory 16, 1992, 51-59) proposed the following conjecture. Conjecture. Let D be a 2-strongly connected digraph of order n such that for all distinct pairs of non-adjacent vertices x, y and w, z, we have d(x)+d(y)+d(w)+d(z) ≥ 4n − 3. Then D is Hamiltonian. In this note, we prove that if D satisfies the conditions of this conjecture, then (i) D has a cycle factor; (ii) If {x, y} is a pair of non-adjacent vertices of D such that d(x) + d(y) ≤ 2n − 2, then D is Hamiltonian if and only if D contains a cycle passing through x and y; (iii) If {x, y} a pair of non-adjacent vertices of D such that d(x)+d(y) ≤ 2n−4, then D contains cycles of all lengths 3, 4, . . . , n−1; (iv) D contains a cycle of length at least n − 1

Bispindles in Strongly Connected Digraphs with Large Chromatic Number

A $(k_1+k_2)$-bispindle is the union of $k_1$ $(x,y)$-dipaths and $k_2$ $(y,x)$-dipaths, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. showed that for every $(1,1)$- bispindle $B$, there exists an integer $k$ such that every strongly connected digraph with chromatic number greater than $k$ contains a subdivision of $B$. We investigate generalizations of this result by first showing constructions of strongly connected digraphs with large chromatic number without any $(3,0)$-bispindle or $(2,2)$-bispindle. We then consider $(2,1)$-bispindles. Let $B(k_1,k_2;k_3)$ denote the $(2,1)$-bispindle formed by three internally disjoint dipaths between two vertices $x,y$, two $(x,y)$-dipaths, one of length $k_1$ and the other of length $k_2$, and one $(y,x)$-dipath of length $k_3$. We conjecture that for any positive integers $k_1, k_2,k_3$, there is an integer $g(k_1,k_2,k_3)$ such that every strongly connected digraph with chromatic number greater than $g(k_1,k_2,k_3)$ contains a subdivision of $B(k_1,k_2;k_3)$. As evidence, we prove this conjecture for $k_2=1$ (and $k_1, k_3$ arbitrary).

Bounds on the domination number of a digraph and its reverse

Let D be a digraph. A dominating set of D is the subset S of V(D) such that each vertex in V(D)?S is an out-neighbor of a vertex in S. The minimum cardinality of a dominating set of G is denoted by ?(D). We let D?denote the reverse of D. In [Discrete Math. 197/198 (1999) 179-183], Chartrand, Harary and Yue proved that every connected digraph D of order n ? 2 satisfies ?(D)+ ?(D?) ? 4n 3 and characterized the digraphs D attaining the equality. In this paper, we pose a reduction of the determining problem for (D)+(D?) using the total domination concept. As a corollary of such a reduction and known results, we give new bounds for (D)+(D?) and an alternative proof of Chartrand-Harary-Yue theorem.

Restricted connectivity of total digraph

For a strongly connected digraph [Formula: see text], an arc-cut [Formula: see text] is a [Formula: see text]-restricted arc-cut of [Formula: see text] if [Formula: see text] has at least two strong components of order at least [Formula: see text]. The [Formula: see text]-restricted arc-connectivity [Formula: see text] is the minimum cardinality of all [Formula: see text]-restricted arc-cuts. The [Formula: see text]-restricted vertex-connectivity [Formula: see text] can be defined similarly. In this paper, we provide upper and lower bounds for [Formula: see text] and [Formula: see text] for the total digraph [Formula: see text] of [Formula: see text].

Vertex-Transitive Digraphs of Order $p^5$ are Hamiltonian

We prove that connected vertex-transitive digraphs of order $p^{5}$ (where $p$ is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup $G$ of prime power order such that $G'$ is generated by two elements or elementary abelian is Hamiltonian.