Advanced Vertex Degrees and Graph Properties Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) In a simple undirected graph, the degree of a vertex (v) is:

Explanation

In a simple graph, the degree equals the number of vertices adjacent to (v).

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About This Quiz
Advanced Vertex Degrees And Graph Properties Quiz - Quiz

Think you can infer a graph’s structure just from its degree sequence? This graduate-level quiz challenges your understanding of degree-based reasoning in simple graphs, multigraphs, and directed graphs. You’ll apply the handshake lemma, test whether sequences are graphical, and explore how vertex degrees limit possible structures. You’ll also work with... see morek-regular graphs, star graphs, and degree constraints in complete graphs, all while sharpening your ability to reason about adjacency and connectivity. By the end, you’ll be confident using degrees to uncover hidden properties and determine what kinds of graphs can or cannot exist. see less

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2) In an undirected graph with 8 vertices and 11 edges, what is the sum of all vertex degrees?

Explanation

Handshake lemma: total degree = 2×11 = 22.

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3) A graph has 6 vertices with degrees 3,3,2,2,2,2. How many edges does it have?

Explanation

Sum of degrees = 14 → edges = 14/2 = 7.

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4) In any simple graph on (n) vertices, the degree of each vertex is at most (n-1).

Explanation

A vertex cannot connect to itself, so max degree is (n−1).

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5) In the complete graph (K_7), what is the sum of degrees of all vertices?

Explanation

Each vertex has degree 6 → total 42.

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6) In a directed graph, the sum of all in-degrees equals:

Explanation

Each directed edge contributes 1 to in-degree.

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7) A vertex is pendant iff degree is:

Explanation

Pendant means degree 1.

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8) A graph is k-regular if every vertex has degree k.

Explanation

Regular graphs have uniform degree.

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9) Which degree sequence cannot occur in a simple graph on 4 vertices?

Explanation

Degree 3 forces all vertices connected → none can have degree 0.

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10) Star graph K1,n: degree of each leaf?

Explanation

Leaves connect only to center.

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11) Multigraph: v has 3 loops, 2 edges. deg(v)?

Explanation

Loops are 2 each: 6+2=8.

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12) Odd-degree vertices count is always even.

Explanation

Handshake lemma yields even total degree.

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13) Simple graph with degrees 4,4,4,4,4: number of vertices?

Explanation

Degree 4 requires ≥5 vertices.

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14) Simple graph 10 vertices, one vertex deg 3: max edges?

Explanation

Remaining 9 form 36 edges +3 =39.

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15) In simple graph on n vertices, at least two vertices share a degree.

Explanation

Pigeonhole principle.

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In a simple undirected graph, the degree of a vertex (v) is:
In an undirected graph with 8 vertices and 11 edges, what is the sum...
A graph has 6 vertices with degrees 3,3,2,2,2,2. How many edges does...
In any simple graph on (n) vertices, the degree of each vertex is at...
In the complete graph (K_7), what is the sum of degrees of all...
In a directed graph, the sum of all in-degrees equals:
A vertex is pendant iff degree is:
A graph is k-regular if every vertex has degree k.
Which degree sequence cannot occur in a simple graph on 4 vertices?
Star graph K1,n: degree of each leaf?
Multigraph: v has 3 loops, 2 edges. deg(v)?
Odd-degree vertices count is always even.
Simple graph with degrees 4,4,4,4,4: number of vertices?
Simple graph 10 vertices, one vertex deg 3: max edges?
In simple graph on n vertices, at least two vertices share a degree.
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