3.(a) Draw C6, C7, We and W7. 3.(b) Are C, and W, regular graphs? Justify your answer. 3.(c) How many edges and vertices do Ce, C7, We and w, have? 3.(d) Let a, be the number of edges of W, and b, be the number of edges of Cn, Find a recurrence relation for a1, a2, az, .. and b, b2, bs, ... . Guess the explicit formula for a, and b. Justify your answer. 3.(e) For which values of n do W, and C, have Euler circuit? Justify your answer. 3.(f) For which values of n does W, have Hamiltonian circuit? If so, label the vertices and find the Hamiltonian circuit. 3.(g) For which values of n do W, and C, are bipartite? Justify your answer.

3.(a) Draw C6, C7, We and W7. 3.(b) Are C, and W, regular graphs? Justify your answer. 3.(c) How many edges and vertices do Ce, C7, We and w, have? 3.(d) Let a, be the number of edges of W, and b, be the number of edges of Cn, Find a recurrence relation for a1, a2, az, .. and b, b2, bs, ... . Guess the explicit formula for a, and b. Justify your answer. 3.(e) For which values of n do W, and C, have Euler circuit? Justify your answer. 3.(f) For which values of n does W, have Hamiltonian circuit? If so, label the vertices and find the Hamiltonian circuit. 3.(g) For which values of n do W, and C, are bipartite? Justify your answer.

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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Question

A simple graph is called regular if every vertex of the graph has the same degree. An
n-regular graph is a graph with every vertex with degree n.
• The graph of cycle denoted by C; for n > 3, consists of n vertices v1, '2,..., e, and
edges {v, v2}, {v2, v3}, ..., {vn-1, t'n} and {t,, t;}.
• The graph of wheel denoted by W, is obtained when an additional vertex is added to
cycle Cn, for n > 3, and connect this new vertex to each of n vertices by new edges.
3.(a) Draw Cę, C7, We and W7.
3.(b) Are C, and W, regular graphs? Justify your answer.
3.(c) How many edges and vertices do Cg, C7, W, and W, have?
3.(d) Let a, be the number of edges of Wn and b, be the number of edges of Cn, Find a recurrence
relation for a1, a2, a3, ... and b1, b2, b3, .... Guess the explicit formula for an and ba. Justify
your answer.
3.(e) For which values of n do W, and C, have Euler circuit? Justify your answer.
3.(f) For which values of n does W, have Hamiltonian circuit? If so, label the vertices and find
the Hamiltonian circuit.
3.(g) For which values of n do W, and C, are bipartite? Justify your answer.

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