Let G be a connected graph with vertices labeled v1 , v2 ,. We will prove the first part of the theorem. The proofs of the other parts are left for you as exercises. The fact that there are no zeros on row j of Sk implies that the distance from vj to any other vertex is at most k. We can use adjacency matrices to create other types of graph-related matrices. The steps given below describe the construction of a new matrix, using the matrix sums Sk defined earlier.
Carefully read through the process, and before you read the paragraph that follows! For each pair i, j, the i, j entry of M is the distance from vi to vj. The matrix M is called the distance matrix of the graph G. Exercises 1. Give the adjacency matrix for each of the following graphs.
Without computing the matrix directly, find A3 where A is the adjacency matrix of K4. If A is the adjacency matrix for the graph G, show that the j, j entry of A2 is the degree of vj. Let A be the adjacency matrix for the graph G. Prove that the number of triangles in G is 16 Tr A3. Find the 1, 5 entry of A where A is the adjacency matrix of C10 and where the vertices of C10 are labeled consecutively around the cycle.
Use Theorem 1. The graph G has adjacency matrix A and distance matrix D. Give the distance matrices for the graphs in Exercise 1. You should create these matrices directly — it is not necessary to use the method described in the section. Graph Theory 1.
In this section we will discuss several models in which the idea of distance is significant. Such discoveries are enjoyable, for sure, but perhaps the frequency with which they occur ought to keep us from being as surprised as we typically are when we experience them. We can get a better feel for this phenomenon by using a graph as a model.
Define the Acquaintance Graph, A, to be the graph where each vertex represents a person, and an edge connects two vertices if the corresponding people know each other. The context here is flexible — one could create this graph for the people living in a certain neighborhood, or the people working in a certain office building, or the people populating a country or the planet.
Might there be a person or a group of people with no connection direct or indirect at all to another group of people? It is, of course, difficult if not impossible to confirm this. For one, A is enormous, and the kind of computation required for confirmation is nontrivial to say the least!
Further, the matrix A is not static — vertices and edges appear all of the time. He randomly selected several hundred people from certain communities in the United States and sent a 3 The graph could be made even larger by allowing the vertices to represent all people, living or dead. Edges disappear when amnesia strikes.
If the recipient knew this target personally, the recipient was to mail the packet directly to him. The new recipient was to follow the same rules: Either send it directly to the target if known personally or send it to someone who has a good chance of knowing the target. Milgram tracked how many steps it took for the packets to reach the target. Of the packets that eventually returned, the median number of steps was 5!
Wow, what a small world! This question, first asked by a group of college students in , was the beginning of what was soon to become a national craze: The Kevin Bacon Game. The object of the game is to connect actors to Bacon through appearances in movies. Since Thompson has not appeared with Bacon in a movie, two steps is the best we can do. We say that Thompson has a Bacon number of 2. Can you sense the underlying graph here? The ease with which Kevin Bacon can be connected to other actors might lead one to conjecture that Bacon is the unique center of Hollywood.
Is this true? На данный момент в продаже есть масса проверенных и дешевых моделей, потому таковая покупка не сильно стукнет по вашему бюджету. В крайнее время резиновые лодки в Крыму пользуются высочайшей популярностью. Для вас требуется лишь разобраться, какая вместительность для вас будет самой хорошей.
Самым увлекательным вариантом числятся модели RIB. Их основное преимущество состоит в том, что на их можно ставить моторы. Правда габариты побольше, что вызывает трудности при перевозке и хранении. Резиновые лодки недорого стоят. Данный материал различается от лодки ПВХ тем, что практически в 2 раза легче. Ежели длина избранной модели меньше 2,5 метров, она способна выдерживать не наиболее килограмм, то для вас не придется регистрировать ее в качестве плавательного средства. Обратите внимание на качество других материалов.