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Graph Theory
Published in Wai-Kai Chen, Mathematics for Circuits and Filters, 2000
A graph G = (V, E) consists of two sets: a finite set V = (v1, v2, …, vn) of elements called vertices and a finite set E = (e1, e2, …, em) of elements called edges. Each edge is identified with a pair of vertices. If the edges of G are identified with ordered pairs of vertices, then G is called a directed or an oriented graph. Otherwise G is called an undirected or a nonoriented graph. Graphs are amenable for pictorial representations. In a pictorial representation each vertex is represented by a dot and each edge is represented by a line segment joining the dots associated with the edge. In directed graphs we assign an orientation or direction to each edge. If the edge is associated with the ordered pair (vi, vj), then this edge is oriented from vi to vj. If an edge e connects vertices vi and vj then it is denoted by e = (vi, vj). In a directed graph (vi, vj) refers to an edge directed from vi to vj. A graph and a directed graph are shown in Fig. 7.1. Unless explicitly stated, the term “graph” may refer to an undirected graph or to a directed graph.
Structure and Representation
Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018
Jonathan L. Gross, Jay Yellen, Mark Anderson
Remark: A generalization of graph isomorphism, called a linear graph mapping, is an adjacency-preserving mapping between the vertex-sets that is not required to preserve non-adjacency and is not necessarily bijective. The reader familiar with group theory will notice this analogy: linear graph mapping is to graph isomorphism as group homomorphism is to group isomorphism. In fact, some authors use the term graph homomorphism instead of linear graph mapping.
Graph Theory
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
Generally we shall use the term ‘graph' without qualification to mean undirected graph. If we need to emphasize a specific graph Γ we will write VΓ, EΓ and δΓ for the sets V and E and the function δ: E → 퓟(V) respectively.
Extremal inverse eigenvalue problem for irreducible acyclic matrices
Published in Applied Mathematics in Science and Engineering, 2022
Debashish Sharma, Bhaba Kumar Sarma
Throughout this paper, the term graph will mean an undirected graph on the vertex set without loops or multiple edges. Let and be two graphs on V. A permutation π of V is an isomorphism of onto , if is an edge in if and only if is an edge in . If such a π exists, and are said to be isomorphic. If , then the isomorphism π is called an automorphism of G. The set of automorphisms of G form a finite group, denoted by .
A review of approaches for topic detection in Twitter
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2021
Zeynab Mottaghinia, Mohammad-Reza Feizi-Derakhshi, Leili Farzinvash, Pedram Salehpour
A hybrid relations analysis approach to integrating semantic information and co-occurrence information among terms for topic detection has been proposed in (Zhang et al., 2016). Specifically, the method fuses multiple relations into a uniform term graph by combining ID theory with the topic modelling method and detects topics from the graph using a graph analytical method. First of all, an Idea Discovery algorithm (Idea-Graph) is adopted to mine co-occurrence relations (especially latent co-occurrence relations) for converting the corpus into a term graph. After that, an LDA-based semantic relations extraction method enriches the graph with semantic information. Finally, a graph analytical method is exploited by the graph for topic detection. Two variants of this approach have been evaluated: (1) IG: it only employs Idea-Graph to generate the term graph, and (2) LDA-IG: it has equipped with Idea-Graph and LDA.
The impact of node arrival process and stochastic edge growth on scale-free distribution in complex networks
Published in International Journal of Modelling and Simulation, 2020
F. Safaei, H. Yeganloo, M. Moudi
We start with a formal definition of the graph as follows. Let be a finite set of unknown elements and let be a set of all ordered pair of V. Each relation defined on the set V is an arbitrary subset so that n = |G| is the size of a graph. The relation E is symmetric if and is reflexive if. Therefore, we able to define a simple graph as G (V, E) where V is a finite set of nodes (vertices) and E is a reflexive relation on the set V and it is known as the link (edge) of graph G. In this paper, we study the simple unweighted non-oriented graphs, which contain no self-loops or multiple edges. When we use the term ‘graph’ we mean a simple graph, otherwise, we point out explicitly. G (n, m) is a set of all graphs that have n nodes and m edges. In a non-oriented graph, the relation E is asymmetric.