Figure 1:
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Figure 2:
Birla Institute of Technology & Science, Pilani
Department of Mathematics
Second Semester 2022-2023
MATH F243
Tutorial-3
Q.1 Prove that if u and v are adjacent vertices then their eccentricities differ by atmost one.
Q.2 Compute the diameter and radius of the Km,n ?
Q.3 Let T be a tree on n vertices that has no vertex of degree 2. Show that T has more than n/2 leaves?
Q.4 Consider a tree T having 50 leaves and equal number of vertices of degree 2, 3, 4 and 5 and it has no vertices of degree greater
than 5. Compute the order of T .
Q.5 Find all trees T such that T C is also a tree.
Q.6 Let T be an n-vertex tree that has exactly 2k vertices of odd degree. Show that T can be split into k edge-disjoint paths (i.e.,
T is the union of k edge-disjoint paths).
Q.7 For any connected graph G, rad(G) ≤ diam(G) ≤ 2rad(G) ?
Q.8 Find radius and diameter of the graphs in Figure 1.
Q.9 Find eccentricity of each of the vertex of the graphs in Figure 2. Hence, find radius and diameter of the given graphs.
Q.10 Let T be a tree of order k. If G is a graph with δ(G) ≥ k − 1, then T is isomorphic to some subgraph of G.
Q.11 Draw a simple connected graph G such that diameter of G and GC is three.