What is the maximum number of edges G could have an still be disconnected… What is the maximum number of edges in a disconnected graph on n vertices from CS 70 at University of California, Berkeley How to enable exception handling on the Arduino Due? In a simple undirected graph with n vertices what is maximum no of edges that you can have keeping the graph disconnected? In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into isolated subgraphs. We consider both "extremes" (the answer by N.S. Beethoven Piano Concerto No. It has n(n-1)/2 edges . Then, each vertex in the first piece has degree at most $k-1$, therefore the number of edges in the first component is at most $\frac{k(k-1)}{2}$, while the number of edges in the second component is at most $\frac{(n-k)(n-k-1)}{2}$. Welcome to math.SE. Alternate solution How many connected graphs over V vertices and E edges? It only takes a minute to sign up. (Equivalently, if any edge of the graph is part of a k -edge cut). you can check the value by putting the different value of x and then you will get "U" type of shape. Then, the minimum number of edges in X is n 1. The maximum number of simple graphs with n=3 vertices −. What is the maximum number of edges possible in this graph? Class 6: Max. To finish the problem, just prove that for $1 \leq k \leq k-1$ we have Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Can you legally move a dead body to preserve it as evidence? What is the possible biggest and the smallest number of edges in a graph with N vertices and K components? A connected graph on $n$ vertices has at least $n-1$ edges, this minimum being attained when the graph is a tree. Therefore our disconnected graph will have only two partions because as number of partition increases number of edges will decrease. Even if it has more than 2 components, you can think about it as having 2 "pieces", not necessarily connected. Replacing the core of a planet with a sun, could that be theoretically possible? Thereore , G1 must have. Let G be a graph with n vertices. Then, each vertex in the first piece has degree at k-1 Celestial Warlock's Radiant Soul: are there any radiant or fire spells? Now, according to Handshaking Lemma, the total number of edges in a connected component of an undirected graph is equal to half of the total sum of the degrees of all of its vertices. 6-20. I think that the smallest is (N-1)K. The biggest one is NK. Therefore our disconnected graph will have only two partions because as number of partition increases number of edges will decrease. Simple, directed graph? This is because instead of counting edges, you can count all the possible pairs of vertices that could be its endpoints. Proof. Proof. Maximum number of edges in a complete graph = nC2. Home Browse by Title Periodicals Discrete Mathematics Vol. Request PDF | Maximum number of edges in a critically k-connected graph | A k-connected graph G is said to be critically k-connected if G−v is not k-connected for any v∈V(G). If they have the same amount, you have $2\binom{n/2}{2}$ edges if $n$ is even, or $\binom{(n-1)/2}{2}+\binom{(n+1)/2}{2}$ if $n$ is odd. The connectivity of a graph is an important measure of its resilience as a network. maximum number of edges in a graph with components. Since we have to find a disconnected graph with maximum number of edges with n vertices. Why does "nslookup -type=mx YAHOO.COMYAHOO.COMOO.COM" return a valid mail exchanger? Does the Pauli exclusion principle apply to one fermion and one antifermion? The maximum number of edges with n=3 vertices −. Here's another way to derive that result, if you happen to know that for any (simple) graph $G,$ either the graph $G$ or its complement $\overline G$ is connected (see this question.) If you add them to your graph, you get a simple graph, which by handshaking lemma, has at most $\frac{n(n-1)}{2}$ edges. a) G is a complete graph b) G is not a connected graph ... What is the maximum number of edges in a bipartite graph having 10 … of edges in a DISCONNECTED simple graph…. If the edge is removed, the graph becomes disconnected… Hence the revised formula for the maximum number of edges in a directed graph: 5. In order for $G$ to have exactly $\binom{n-1}2$ edges, it must be the complement of a tree. rev 2021.1.7.38269, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. @anuragcse15, nice question!! Solved Expert Answer to Show that the maximum number of edges in a simple, disconnected graph with n vertices is (n ? deleted , so the number of edges decreases . What is the minimum number of edges G could have and still be connected? 2)/2. First, note that the maximum number of edges in a graph (connected or not connected) is 1 2 n (n − 1) = (n 2). Is it connected or disconnected? By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. The last remaining question is how many vertices are in each component. Second, for all n ≥ 1, every graph with n vertices and more than m(n) edges is connected.] By Lemma 9, every graph with n vertices and k edges has at least n k components. This can be proved by using the above formulae. Support your maximality claim by an argument. Below is the implementation of the above approach: That's the same as the maximum number of [unique] handshakes among $n$ people. Use MathJax to format equations. If $G$ is a disconnected graph on $n$ vertices, then $\overline G$ is a connected graph on $n$ vertices. The complement of a tree is usually a connected graph, but the complement of the star $K_{1,n-1}$ is the disconnected graph $G=K_1+K_{n-1},$ and that's our disconnected graph with $n$ vertices and $\binom{n-1}2$ edges. 3. Can I print plastic blank space fillers for my service panel? site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Please use Mathjax for better impact and readability, The maximum no. =1/2*(2x2 -2nx + n2 -n),              where , 1<= x <= n-1. Now assume that First partition has x vertices and second partition has (n-x) vertices. Origin of “Good books are the warehouses of ideas”, attributed to H. G. Wells on commemorative £2 coin? Is it normal to need to replace my brakes every few months? Therefore, total number of edges = nC2 - (n-1) = n-1C2. This is because instead of counting edges, you can count all the possible pairs of vertices that could be its endpoints. Maximum number of edges in connected graphs 71 In order for equality to hold here we would have to have n = k + 2 which cannot be since k + 1 -- n /2. It's also worth mentioning that the problem of maximizing the number of edges in a graph forbidding an even cycle of fixed length is well studied (see, e.g., the Bondy-Simonovits Theorem). Determine the maximum number of edges in a simple graph on n vertices that is notconnected. 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