Let G be a connected graph of order 6 with degree sequence (2,2,2,2,2,2). (a) Determine all the

Let G be a connected graph of order 6 with degree sequence...

Key Concepts

Induced Subgraph

An induced subgraph is formed from a graph by selecting a subset of vertices and considering all of the edges between them that appear in the original graph. This concept is crucial when studying the underlying structure of a graph, as induced subgraphs preserve the relationships among the chosen vertices without adding or omitting any connections present in the parent graph.

Spanning Subgraph

A spanning subgraph is a subgraph that includes all the vertices of the original graph but may have only a subset of the edges. Spanning subgraphs are important in various areas of graph theory, and they help analyze how edge removals affect properties like connectivity.

Graph Isomorphism

Graph isomorphism is a concept that determines when two graphs can be considered 'the same' in structure, despite possible differences in vertex labels or drawings. Two graphs are isomorphic if there exists a one-to-one correspondence between their vertices that preserves the adjacency relation. This concept is key when distinguishing nonisomorphic subgraphs, ensuring that only structurally unique graphs are considered.

Degree Sequence

The degree sequence of a graph is the list of vertex degrees usually arranged in non-increasing order. It provides important information about the structure of the graph, such as regularity. In the context of the problem, having a degree sequence where every vertex has degree 2 indicates that each vertex connects to exactly two others, leading to a cycle in a connected graph.

Connected Graph

A connected graph is one in which there is a path between every pair of vertices. This property ensures that the graph is in one piece, meaning no vertex is isolated or in a disconnected component. In problems involving subgraphs, understanding which parts retain connectivity is often crucial.

Cycle Graph

A cycle graph is a graph that forms a single closed loop in which each vertex is connected to exactly two other vertices. It is denoted by C_n where n represents the number of vertices. In a cycle graph of order 6, every vertex has degree 2, which is a central fact in understanding the structure of the original graph described in the problem.

Graph

A graph is a mathematical structure used to model pairwise relations between objects. It consists of vertices (or nodes) and edges connecting pairs of vertices. Graphs can be used to represent networks, relationships, and many other structures, and are studied using various properties such as connectivity, cycles, paths, and degree sequences.

Transcript

00:01 We were asked to find the number of non -isomorphic simple graphs with six vertices, where the degree of each vertex is three.

00:22 So i'll call these vertices a, b, c, t, e, and f.

00:38 And because these are simple graphs, we have by the handshaking theorem that two times the number of edges is going to be equal to the number of vertices six times the degree of each vertex, which is three.

00:54 So we have the number of edges is going to be nine.

01:02 So this means that if we have nine edges, it's consider the possibilities.

01:36 So we could have that a connects to b and b connects to c, and c connects to d, d connects to e, e connects to f, f connects to a.

01:51 So for all the vertices have a degree of two.

01:55 However, you've only used one, two, three, four, five, six edges.

02:01 So we have three edges left.

02:02 You want to make it so that each vertex has a degree of three.

02:09 So to do this, simply connect across the graph.

02:17 So have an edge with a and d as well as an edge with f and c and an edge with e and v.

02:28 This is a clearly valid construction here.

02:33 Another possibility.

02:50 So we'll have the same outer edges, and a is still connected to d.

02:57 However, instead of connecting f to c, let's connect f to b and connect e to c...

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