Answered: 5. If G is a loop-free undirected graph… | bartleby

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 36EQ

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5. If G is a loop-free undirected graph with at least one edge, prove that G is bipartite if and
only if χ(G) = 2.

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Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph obtained from G by creating a new edge, e, joining v and w. Prove that x(G) = min{x(G+e), x((G+e) 4e)}.
Let G = (V, E) be a finite bipartite graph with bipartition (A, B) where |A| = |B|. We say that G satisfies the marriage condition iff for every SCA, ING(S) ≥ |S| where NG(S) = {be B: (3a € S)({a, b} = E)} It should be clear that if G has a perfect matching, then it satisfies the marriage condition. Hall's theorem says that the converse is also true.
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5. If G is a simple graph with d(v) 2 k, Vv € V (G), then G contains a path of length at least k. If k2 2, then G contains a cycle of length k+1.
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7. Suppose a graph G = (V, E) has n vertices and m edges (i.e., |V| = n and |E| = m). Further suppose that m n. (b) have degree exactly 1. Prove that there are at least 2(n – m) vertices which
= 2 and let G₁ = (V, E) be the graph with vertex set V i+1}:1
(i) Let e1, e2, . . . , one connection of the graph G and let xi−1 and xi be the sites of the connection ei (1 ≤ i ≤ n). Show that every closed walk (e1, e2, . . . , en) is of length at least 3 with pairs of distinct pointsx1, x2, . . . , xn cycles.(ii) A graph containing no cycles is called acyclic. A walk is acyclic if the subgraph consisting of points and links of the walk is acyclic. Prove: a walk has all distinct points if and only if it is an acyclic sequence.(iii) If and are in different points of the graph G and if there is a walk in G from u to v, show that then there is an acyclic sequence from u to v.(Explain precisely that every shortest walk from u to v is actually an acyclic path.)
3. A simple graph H is the square of a simple graph G if V (H) = V(G) and xy is an edge in H if and only if x # y and the distance between x and y in G is at most two. Prove that for n ≥ 4, the square, C2, of the cycle C₁ is planar if and only if n is even.
8. Answer these two questions:
6. (In this problem, y denotes the domination number of a graph.) Let G be a graph such that y(G) > 3. Prove that diam(G) < 2.

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