# Graph Theory

**Topics:**Graph theory, Planar graph, Complete bipartite graph

**Pages:**25 (6861 words)

**Published:**September 16, 2013

T. Britz/D. Chan/D. Trenerry

§5 Graph Theory

Loosely speaking, a graph is a set of dots and dot-connecting lines. The dots are called vertices and the lines are called edges. Formally, a (ﬁnite) graph G consists of A ﬁnite set V whose elements are called the vertices of G; A ﬁnite set E whose elements are called the edges of G; A function that assigns to each edge e ∈ E an unordered pair of vertices called the endpoints of e. This function is called the edge-endpoint function. Note that these graphs are not related to graphs of functions. Graphs can be used as mathematical models for networks such roads, airline routes, electrical systems, social networks, biological systems and so on. Graph theory is the study of graphs as mathematical objects. 1

Exercise. Consider the following graph G with vertices and edges V = {v1 , v2 , v3 , v4 , v5 } and E = {e1 , e2 , e3 , e4 , e5 , e6 , e7 } :

v3 e2 e5 v1 e 1 v2 e3 v4 e 6 v5 e7 e4

Edge e1 e2 e3 e4 e5 e6 e7

Endpoints {v1 , v2 } {v2 , v3 } {v2 , v4 } {v3 , v4 } {v3 , v4 } {v4 , v5 } {v5 }

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Example. Below are 3 diﬀerent pictorial representations of the same graph. e1 v1 e2 v2 e1 e3 v3 e4 v1 e4 e2 e3 v3 v2 e4 v1 e1 e3 v3 e2 v2

The edge-endpoint function of this graph is as follows: Edge e1 e2 e3 e4 Endpoints {v1 , v2 } {v1 , v2 } {v2 , v3 } {v3 }

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If the edge e ∈ E has endpoints v, w ∈ V , then we say that the edge e connects the vertices v and w; the edge e is incident with the vertices v and w; the vertices v and w are the endpoints of the edge e; the vertices v and w are adjacent; the vertices v and w are neighbours. Two edges with the same endpoints are multiple or parallel . A loop is an edge that connects a vertex to the same vertex. The degree of a vertex v, denoted by deg(v), is the number of edges incident with v, counting any loops twice. An isolated vertex is one with degree 0, and a pendant vertex is one with degree 1.

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Exercise. v3 e2 e5 v1 e1 v2 e3 v4 e 6 v5 e7 e4 v6 Vertex v1 v2 v3 v4 v5 v6 Degree 1 3 3 4 3 0

In the diagram, e3 connects vertices v2 and v4 ; v2 and v3 are adjacent/neighbours; e7 is a loop; e4 and e5 are multiple/parallel edges; v1 is a pendant vertex; v6 is an isolated vertex. 5

The Handshaking Theorem. The total degree of a graph is twice the number of edges: 2|E| = v∈V

deg(v) .

Proof.

Each edge has two endpoints and must contribute 2 to the sum of degrees, which is why we count a loop twice.

Example. v3 e2 e5 v1 e 1 v2 e3 v4 e 6 v5 e7

v∈V

v6 e4 2|E| = 2 · 7 = 14 deg(v) = 1 + 3 + 3 + 4 + 3 + 0 = 14

By the Handshaking Theorem, the total degree of a graph must be even. 6

Example. No graph can have vertex degrees 3,3,3,2,2. Why? 3 + 3 + 3 + 2 + 2 = 13 is odd. A simple graph is a graph with no loops or parallel edges. Example.

A simple graph

Not a simple graph

Not a simple graph

Note: each vertex in a simple graph on n vertices has degree at most n − 1. Why? Let v be a vertex. There is no loop at v. No parallel edges =⇒ at most 1 edge connects v to each of the other n − 1 vertices. In total, there are at most n − 1 edges incident on v.

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Exercise. Prove that no simple graph can have the following vertex degrees: 5,4,3,2,2; A simple graph with 5 vertices has all vertex degrees ≤ 4 4,3,3,1,1. Answer Proof by contradiction. Suppose there is a simple graph with vertex degrees 4, 3, 3, 1, 1. Label the corresponding vertices v1 , v2 , v3 , v4 , v5 . deg(v1 ) = 4 =⇒ v1 is adjacent to all the other 4 vertices. deg(v2 ) = 3 =⇒ v2 is adjacent to either v4 or v5 . Without loss of generality, suppose there’s an edge connecting v2 to v4 . Then v4 is adjacent to both v1 and v2 contradicting deg(v4 ) = 1.

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SOME NAMED GRAPHS 1. The complete graph Kn (n ≥ 1) is a simple graph with n vertices; exactly one edge between each pair of distinct vertices. Hence Kn has C(n, 2) edges.

K1

K2

K3

9

K4

K5

K6

2. The cyclic graph Cn (n...

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