Core Concepts in Graph Theory
Medium-difficulty multiple-choice quiz on undirected, complete, regular, self-loop, multigraph, and proper (simple) graph concepts and consequences.
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Quiz Questions & Answers
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Question 1: Which property defines an undirected graph?
Each edge has no orientation; edges connect vertices symmetrically.
Every vertex must have the same degree.
Multiple parallel edges and loops are forbidden by definition.
Edges carry a direction from one vertex to another.
Question 2: What makes a graph 'complete'?
Edges are partitioned into disjoint cycles covering all vertices.
No vertex is adjacent to more than one other vertex.
Every pair of distinct vertices is connected by exactly one edge.
Graph contains at least one self-loop at each vertex.
Question 3: Which definition best captures a k-regular graph?
Exactly k vertices have self-loops.
There are k parallel edges between each vertex pair.
Graph has k connected components.
Every vertex has degree exactly k.
Question 4: How does a self-loop affect the degree of its incident vertex in standard undirected graph conventions?
It reduces the degree by one due to redundancy.
It contributes one to the vertex degree.
It contributes two to the vertex degree.
Self-loops are ignored when computing degree.
Question 5: Which statement distinguishes a multigraph from a simple (proper) graph?
A multigraph allows multiple parallel edges between the same vertices; a simple graph forbids them.
A multigraph requires every vertex to have the same degree.
A multigraph has weighted edges but simple graphs do not.
A multigraph is always directed, while a simple graph is undirected.
Question 6: Why can't a simple complete graph of n vertices include self-loops and still be called simple?
Because loops change vertex colors in proper colorings.
Because simple graphs prohibit self-loops and allow exactly one edge between distinct vertices only.
Because loops force the graph to be disconnected.
Because loops would reduce the total number of edges below maximum.
Question 7: Which consequence follows if every vertex in a finite undirected graph has even degree?
The graph must contain at least one self-loop.
The graph must be complete.
The graph's edges can be partitioned into cycles (Eulerian condition for component-wise traversal).
The graph has no cycles at all.
Question 8: Scenario: You need a data model allowing parallel connections and occasional self-connections between entities. Which graph type fits best?
Complete graph, because it ensures every pair is linked once.
Regular graph, because all vertices must have same degree.
Multigraph, because it permits parallel edges and can include self-loops when needed.
Simple (proper) graph, because it restricts edges for clarity.