Algebraic Graph Theory

Godsil, Chris; Royle, Gordon F.

doi:10.1007/978-1-4613-0163-9

Cited by 5,401 publications

(4,447 citation statements)

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“…We show that there exists a smooth orientation (of the incidence graphs of projective planes) that has the required strong element connectivity. It may be possible to extend these results to other families of hypergraphs of interest in algebraic graph theory [8]; perhaps, some new general techniques could be developed.…”

Section: Our Resultsmentioning

confidence: 99%

On orienting graphs for connectivity: Projective planes and Halin graphs

Cheriyan

Zou²

2012

Operations Research Letters

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Nash-Williams proved that the edges of a k-edge connected (undirected) graph can be oriented such that the resulting directed graph is ⌊ k 2 ⌋-edge connected. A long-standing goal in the area is to obtain analogous results for other types of connectivity, such as node connectivity, element connectivity, and hypergraph edge connectivity.We focus on two special classes of graphs, namely, incidence graphs of projective planes and (generalized) Halin graphs, and we prove some analogs of Nash-Williams' result for these classes.

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Section: Our Resultsmentioning

confidence: 99%

On orienting graphs for connectivity: Projective planes and Halin graphs

Cheriyan

Zou²

2012

Operations Research Letters

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“…Notice that the structure matrix defined above is similar to the wellknown adjacency matrix in graph theory; see [10]. However, unlike the adjacency matrix, the diagonal entries in the structure matrix will be assumed to be 1.…”

Section: Problem Formulation and Preliminariesmentioning

confidence: 99%

ℋ<inf>2</inf>-based optimal sparse sliding mode control for networked control systems

Argha

et al. 2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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This paper is devoted to the problem of designing a sparsely distributed sliding mode control for networked systems. Indeed, this note employs a distributed sliding mode control framework by exploiting (some of) other subsystems' information to improve the performance of each local controller so that it can widen the applicability region of the given scheme. To do so, different from the traditional schemes in the literature, a novel approach is proposed to design the sliding surface, in which the level of required control effort is taken into account during the sliding surface design based on the H 2 control. We then use this novel scheme to provide an innovative less-complex procedure that explores sparse control networks to satisfy the underlying control objective. Besides, the proposed scheme to design the sliding surface makes it possible to avoid unbounded growth of control effort during the sparsification of the control network structure. Illustrative examples are presented to show the effectiveness of the proposed approach. keywords Networked control systems, H 2 -based optimal sparse sliding mode control, distributed control systems, linear matrix inequality.

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“…v are adjacent to 4 v and suppose, they are adjacent to 4 v by the edges of order 23 , nn and 1 n respectively as shown in the figure 3.5. ,, n n n up to symmetry.…”

Section: Corollary 33: In a Coxeter Andreev's Tetrahedron T The Ormentioning

confidence: 99%