# CPTS553 Assignment 4 Solved

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1. Recall that the adjacency matrix of a simple graph πΊ with vertex set

{π£1, π£2, β¦ , π£π} is the π Γ π matrix π΄ with entries

π΄π,π = {1Β Β Β Β Β  π£π is adjacent to π£π

0Β Β Β Β Β Β Β Β Β Β Β Β  otherwise

1. Let πΎ3,4 be the complete bipartite graph with vertices

{π£1, π£2, π£3, π£4, π£5,π£6, π£7} and where vertices π£π and π£π are adjacent if and only if π and π have different parity (one of π or π is odd and the other is even.)Β  What does the adjacency matrix π΄ look like in this case?

1. Let πΎ3,4 be the complete bipartite graph with vertices

{π£1, π£2, π£3, π£4, π£5,π£6, π£7} and where vertices π£π and π£π are adjacent if and only if (π β€ 3 and π β₯ 4) orΒ  (π β₯ 4 and π β€ 3).Β  What does the adjacency matrix π΄ look like in this case?

1. We let πΊ be a connected graph. For any vertex π£ Β π, define its eccentricity by the formula ecc.

This is the length of βlongest among all shortest paths with π£ as an endpoint.β

1. Let πΊ be the graph drawn below. Label each vertex with its eccentricity.

1. The diameter of a graph is the maximum among the eccentricities of its vertices and the radius of a graph is the minimum among the eccentricities of its vertices. For the graph πΊ drawn in part a, what is its diameter and radius?
2. A central vertex is a vertex π£ such that ecc(π£) = radius(πΊ). Which of the vertices in the graph πΊ are central vertices?
3. A peripheral vertex is a vertex π£ such that ecc(π£) = diameter(πΊ). Which of the vertices in graph πΊ are peripheral vertices?
4. Explain why it is important for these definitions that πΊ be a connected graph.

One inequality is quite easy and the second can be handled using a central vertex and the triangle inequality.

1. Recall that a bridge is an edge whose deletion increases the number of components of a graph. Also, a link is another term for βnon-bridge.β

1. In the graph πΊ (same as in problem 2a) below, which edges are bridges and which edges are links?

1. If you delete all of the bridges, how many components remain?