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State whether the following statements are true or false. Justify your answers with a short proof or a counter-example i) Every even graph is regular. ii) The independence number of a cycle is equal to its matching number. jii) The diameter of K. . is 2 for all m and n.

State whether the following statements are true or false. Justify your answers with a short proof or a counter-example i) Every even graph is regular. ii) The independence number of a cycle is equal to its matching number. jii) The diameter of K. . is 2 for all m and n.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1) State whether the following statements are true or false. Justify your answers with a short proof or a counter-example i) Every even graph is regular. ii) The independence number of a cycle is equal to its matching number. In. If v is a leaf of a graph and u is the vertex adjacent to v, then ɛ(v) = E(u)+1. iii) The diameter of Km.n is 2 for all m and n. iv) v) Every path is a trail. vi) The diameter of every tree is equal to the twice of the radius of the tree. vii) The number of pairwise nonisomorphic trees on 4 vertices is 8. viii) If G is a 2-edge connected graph, then G is 2-connected. ix) If a graph G has no vertex of degree 5 or more, then x(G) < 4. x) Every subgraph of a planar graph is planar.
Transcribed Image Text:1) State whether the following statements are true or false. Justify your answers with a short proof or a counter-example i) Every even graph is regular. ii) The independence number of a cycle is equal to its matching number. In. If v is a leaf of a graph and u is the vertex adjacent to v, then ɛ(v) = E(u)+1. iii) The diameter of Km.n is 2 for all m and n. iv) v) Every path is a trail. vi) The diameter of every tree is equal to the twice of the radius of the tree. vii) The number of pairwise nonisomorphic trees on 4 vertices is 8. viii) If G is a 2-edge connected graph, then G is 2-connected. ix) If a graph G has no vertex of degree 5 or more, then x(G) < 4. x) Every subgraph of a planar graph is planar.
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