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14 pages, 306 KiB

Open AccessArticle

The λ-Fold Spectrum Problem for the Orientations of the Eight-Cycle

by Şafak Durukan-Odabaşı and Uğur Odabaşı

Symmetry 2023, 15(10), 1930; https://doi.org/10.3390/sym15101930 - 18 Oct 2023

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A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is

^{λ} K_{v}^{*}

, the

λ

-fold complete symmetric digraph on v vertices, that is, the digraph with

λ

directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which

^{λ} K_{v}^{*}

has a D-decomposition is called the

λ

-fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The

λ

-fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the

λ

-fold spectrum problem for each of the remaining 21 orientations. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

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17 pages, 2939 KiB

Open AccessArticle

Ordered-Bipartite Consensus of Multi-Agent Systems under Finite Time Control

by Tiehui Zhang, Xinru Ma, Hengyu Li, Zhaoyan Wang, Shaorong Xie and Jun Luo

Appl. Sci. 2022, 12(23), 12337; https://doi.org/10.3390/app122312337 - 2 Dec 2022

Cited by 2 | Viewed by 1434

Abstract

Since group consensus (GC) control accounts for one-half of the multi-robot coordination domain, as does complete consensus control, this paper will formulate a certain kind of GC and group bipartite consensus (GBC) in finite time for multi-agent systems (MAS). It should be noted that the key to this work is that information transfer under acyclic partition networks is zero-sum between groups and the cooperative–competitive interaction. The general GC will firstly be achieved under acyclic partition, which reflects the fact that such topology allows the MAS to reach agreement in an ordered and hierarchical process. Further, the information communication is zero-sum within each group under the acyclic partition signed digraph. Thus, the group–bipartite agreement for MAS is given under acyclic bipartite topology. These results display not only an ordered, hierarchical state, but also an intra-group symmetric state. In addition, simulation examples are presented in parallel with the theoretical results. Full article

(This article belongs to the Section Computing and Artificial Intelligence)

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