Be sure to do all your work on separate paper, and include all steps where appro

Question

Be sure to do all your work on separate paper, and include all steps where appropriate All homework must follow the formatting rules posted on Blackboard. 1. Could the matrix below be the distance matrix of some graph? Explain 21 0 2 1 242 a 3 2 1 3 2 For G given below, (a) find the radius and diameter of G (b) find the center and periphery of G ic) identify, if possible, any mutually eccentric vertices in G 3. A geodesie joining a vertex in the center of a graph to one of its eccentrie vertioes is called a radial path. Identify a radial path in G from question 2. A pair of vertices weVG) ach that d, diam/(G) is called diametral. A geodesic joining two diametral vertices is called a diametral path Identify a diametral path in G from question 2 Lat H be a spanning sulgraph of graph G. Given vertice.. and e inG, show that dada,r)2dola. S. Suppose radiG)-4, what are the possible values of diam G? Explain why the center of C. is VIC.A graph with this property is called self-eentered. (Really Find two more classes of selentered graphs. 7 Find the center and the centroid of the tree given below 8 Give an example of (a) a tree with center consisting of two vertices and centroid also consisting o two vertices where the center and centroid are disjoit (b) a tree with twe central vertices, beth of which are alse centroidal vertices

Explanation / Answer

1) The given matrix could be a distance matrix for some graph.

Since the elements along diagonal of the matrix represent the distance of vertices from themselves only which is zero and hence justified in the given matrix. This is a 5 * 5 matrix which represent 5 vertex graph.

The given matrix is symmetric hence the matrix is equal to the transpose of the given matrix which is again a condition for distance matrix.

For instance the element a12 in the given graph is equal to 1 .After taking transpose the matrix the element a12 becomes a21 which is also equal to 1 and they represent the same distance between two points back and forth.

Hence the given matrix could be a disatnce matrix.

Thank you !

post the other question separately

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