Mathematical Foundations of Computer Science or Graph Theory
Mathematical Foundations of Computer Science explains the fundamental concepts in mathematics. It can be used by the students in computer science as an introduction to the underlying ideas of mathematics for computer science. It explains topics like mathematical logic, predicates, relations, functions, combinatorics, algebraic structures and graph theory.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures.
Graph:
In one restricted but very common sense of the term,a graph is an ordered pair G = ( V , E ) comprising:
In the edge {x,y} the vertices x and y are called the endpoints of the edge. The edge is said to join x and y and to be incident on x and y. A vertex may exist in a graph and not belong to an edge. Multiple edges, not allowed under the definition above, are two or more edges that join the same two vertices.
In one more general sense of the term allowing multiple edges, a graph is an ordered triple G = (V,E,ϕ) comprising:
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures.
Graph:
In one restricted but very common sense of the term,a graph is an ordered pair G = ( V , E ) comprising:
- V, a set of vertices (also called nodes or points);
- E ⊆ {{x,y} | x ,y ∈ V and x ≠ y}, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with two distinct vertices).
In the edge {x,y} the vertices x and y are called the endpoints of the edge. The edge is said to join x and y and to be incident on x and y. A vertex may exist in a graph and not belong to an edge. Multiple edges, not allowed under the definition above, are two or more edges that join the same two vertices.
In one more general sense of the term allowing multiple edges, a graph is an ordered triple G = (V,E,ϕ) comprising:
- V, a set of vertices (also called nodes or points);
- E, a set of edges (also called links or lines);
- ϕ : E → {{x,y} | x,y ∈ V and x≠y}, an incidence function mapping every edge to an unordered pair of vertices (that is, an edge is associated with two distinct vertices).
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