Fundamental Concepts of Graph Theory: Subgraphs and Connectivity

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In graph theory, what is an isomorphism between two simple graphs G and H?

A bijection f:V(G)→V(H) such that uv is an edge in G if and only if f(u)f(v) is an edge in H (correct)

A function that maps the vertices of G to the edges of H

A function that maps the edges of G to the edges of H

A function that maps the vertices of G to the vertices of H

What is a complete graph in graph theory?

A graph where each vertex has a different degree

A graph where every vertex has a self-loop

A graph where no two vertices share an edge

A graph where every vertex is connected to every other vertex (correct)

What is the Petersen Graph?

A graph with 5 vertices connected in a cycle

A graph with 10 vertices arranged in a pentagon shape

A graph composed of only complete bipartite graphs

A graph where the vertices are subsets of a 5-element set and edges are pairs of disjoint 2-element subsets (correct)

What is an adjacency matrix in graph theory?

A matrix showing which vertices are adjacent in a graph Signup and view all the answers

What does the degree of a vertex in a graph represent?

The number of edges incident on that vertex Signup and view all the answers

What does an incidence matrix in graph theory typically represent?

The relationship between vertices and edges in a graph Signup and view all the answers

What is a subgraph of a graph?

A graph with the same vertices and edges as the original graph Signup and view all the answers

In graph theory, what does it mean for two vertices to be 'adjacent'?

They share a common endpoint Signup and view all the answers

What do we call the number of edges incident upon a vertex?

Degree Signup and view all the answers

In an adjacency matrix of a graph, what does an entry of '0' represent?

The vertices are not connected Signup and view all the answers

How is the incidence matrix of a graph defined?

A matrix showing which vertices are endpoints of which edges Signup and view all the answers

What does it mean for a graph to be 'isomorphic' to another?

The graphs have the same degree sequence Signup and view all the answers

In graph theory, what is an induced subgraph?

A subgraph formed by a subset of vertices of the original graph Signup and view all the answers

Which of the following sets of vertices forms an independent set in a graph?

{C, D} Signup and view all the answers

What is the defining characteristic of a cut-edge in a graph?

It does not belong to any cycles Signup and view all the answers

Which theorem characterizes a cut-edge based on its presence in cycles?

Theorem 1.2.14 Signup and view all the answers

If a graph contains a cycle where edge e=(x, y), how does this imply the connectivity of the component containing e?

The component contains a path from x to y Signup and view all the answers

How is an induced subgraph different from a regular subgraph?

An induced subgraph is defined by the subset of vertices and their connections Signup and view all the answers