These notes have not been checked by Prof. A.G. Thomason and should not be regarded as ocial notes for the course. In particular, the responsibility for any errors is mine please email Sebastian Pancratz (sfp25) with any comments or corrections.

This book contains the successful invited submissions [1–10] to a special issue of Symmetry on the subject area of ‘graph theory’. Although symmetry has always played an important role in graph theory, in recent years, this role has increased significantly in several branches of this field, including, but not limited to: Gromov hyperbolic graphs, metric dimension of graphs, domination theory, a...

This is a replacement paper. Some of the conclusions from statistical graph theory are reiterated in the sixth section, but not fully resolved therein. Most of the redundancies in the later sections have been deleted. An important number-theoretical technique in the opening section relates to one of the main techniques in the last (sixth) section.

This development provides a formalization of directed graphs, supporting (labelled) multi-edges and infinite graphs. A polymorphic edge type allows edges to be treated as pairs of vertices, if multi-edges are not required. Formalized properties are i.a. walks (and related concepts), connectedness and subgraphs and basic properties of isomorphisms. This formalization is used to prove characteriz...

MOMENT2 is an algebraic model management framework that permits manipulating models in the Eclipse Modeling Framework (EMF) [1]. Our goal consists in using OMG standards, such as Meta-Object Facility (MOF) [2], Object Constraint Language [3] and Query/View/Transformation (QVT) [4], as a clean interface between formal methods and model-based industrial tools that permits taking the best benefit ...

<p style='text-indent:20px;'>In the framework of fixed topology and stochastic switching topologies, we study mean-square bounded consensus(MSBC) double-integrator multi-agent systems(SMASs) including additive system noises communication noises. Combining algebra, graph theory random analysis, obtain several equivalent conditions for SMASs to reach MSBC. In addition, simulation examples a...

The first geometric-arithmetic index was introduced in the chemical theory as the summation of 2 du dv /(du  dv ) overall edges of the graph, where du stand for the degree of the vertex u. In this paper we give the expressions for computing the first geometric-arithmetic index of hexagonal systems and phenylenes and present new method for describing hexagonal system by corresponding a simple g...