/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Let \(n \in \mathbf{Z}^{+}\)with... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 6

Let \(n \in \mathbf{Z}^{+}\)with \(n \geq 4\). How many subgraphs of \(K_{n}\) are isomorphic to the complete bipartite graph \(K_{1,3}\) ?

Short Answer

Expert verified
The number of subgraphs of \(K_{n}\) that are isomorphic to \(K_{1,3}\) is \(C(n,4) \times C(4,1)\).

Step by step solution

01

Understanding the Concept of Bipartite Graph

A complete bipartite graph \(K_{m,n}\) is a graph in which vertices can be divided into two disjoint sets \(V_{m}\) and \(V_{n}\), each set containing \(m\) and \(n\) vertices respectively, such that there is an edge between every pair of vertices from different sets, that is, every vertex of \(V_{m}\) is connected to every vertex of \(V_{n}\), but there are no edges within the sets. For \(K_{1,3}\), there is one vertex in one set and three in the other set.
02

Generating Subgraphs Isomorphous to \(K_{1,3}\)

To generate such subgraphs from a complete graph \(K_{n}\), one needs to choose 1 vertex from the \(n\) vertices to form one set and 3 others to form the second set for bipartite seperation. Therefore, the problem reduces to the combination of choosing 4 vertices from \(n\) vertices and then choosing 1 vertex from the selected 4 vertices to separate it from the remaining 3.
03

Calculating the Number of Subgraphs

The number of these subgraphs would therefore be the number of ways to choose 4 vertices out of \(n\), multiplied by the number of ways to choose 1 vertex out of the chosen 4 vertices to be distinguished. This is given by: \[C(n,4) \times C(4,1)\] where \(C(n,k)\) denotes the number of combinations of \(n\) objects taken \(k\) at a time.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(G=(V, E)\) be an undirected graph. Define a relation \(\Re\) on \(V\) by \(a \Re b\) if \(a=b\) or if there is a path in \(G\) from \(a\) to \(b\). Prove that \(\Re\) is an equivalence relation. Describe the partition of \(V\) induced by \(\AA\).

a) If \(G=(V, E)\) is a loop-free 3 -regular connected graph with eight vertices, what is \(|E| ?\) b) Find two graphs \(G_{1}\) and \(G_{2}\) that satisfy the conditions in part (a) with \(G_{1}\) planar and \(G_{2}\) nonplanar.

a) Let \(G\) be an undirected graph with \(n\) vertices. If \(G\) is isomorphic to its own complement \(\bar{G}\), how many edges must \(G\) have? (Such a graph is called self-complementary.) b) Find an example of a self-complementary graph on four vertices and one on five vertices. c) If \(G\) is a self-complementary graph on \(n\) vertices, where \(n>1\), prove that \(n=4 k\) or \(n=4 k+1\), for some \(k \in \mathbf{Z}^{+} .\)

a) At the J. \& J. Chemical Company, Jeannette has received three shipments that contain a total of seven different chemicals. Furthermore, the nature of these chemicals is such that for all \(1 \leq i \leq 5\), chemical \(i\) cannot be stored in the same storage compartment as chemical \(i+1\) or chemical \(i+2 .\) Determine the smallest number of separate storage compartments that Jeannette will need to safely store these seven chemicals. b) Suppose that in addition to the conditions in part (a), the following four pairs of these same seven chemicals also require separate storage compartments: 1 and 4,2 and 5,2 and 6, and 3 and 6 . What is the smallest number of storage compartments that Jeannette now needs to safely store the seven chemicals?

Seven towns \(a, b, c, d, e, f\), and \(g\) are connected by a system of highways as follows: (1) I-22 goes from \(a\) to \(c\), passing through \(b ;(2)\) I-33 goes from \(c\) to \(d\) and then passes through\(b\) as it continues to \(f ;(3)\) I-44 goes from \(d\) through \(e\) to \(a ;(4)\) I- 55 goes from \(f\) to \(b\), passing through \(g\); and, (5) I-66 goes from \(g\) to \(d\). a) Using vertices for towns and directed edges for segments of highways between towns, draw a directed graph that models this situation. b) List the paths from \(g\) to \(a\). c) What is the smallest number of highway segments that would have to be closed down in order for travel from \(b\) to \(d\) to be disrupted? d) Is it possible to leave town \(c\) and return there, visiting each of the other towns only once? e) What is the answer to part (d) if we are not required to return to \(c\) ? f) Is it possible to start at some town and drive over each of these highways exactly once? (You are allowed to visit a town more than once, and you need not return to the town from which you started.)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.