maximum number of edges in a disconnected graph

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Let $k$ and $n-k$ be the number of vertices in the two pieces. That's the same as the maximum … It is minimally k -edge-connected if it loses this property when any edges are deleted. Is it normal to need to replace my brakes every few months? Let in the k_{1} component there are m vertices and component k_{2} has p vertices. Can you please explain why it would be maximum at extreme ends... Also please explain why you have subtracted  nC2-(n-1)...? @anuragcse15, nice question!! Replacing the core of a planet with a sun, could that be theoretically possible? So, there is a net gain in the number of edges. What is the possible biggest and the smallest number of edges in a graph with N vertices and K components? Crack in paint seems to slowly getting longer. How can there be a custom which creates Nosar? The connectivity of a graph is an important measure of its resilience as a network. A connected graph on $n$ vertices has at least $n-1$ edges, this minimum being attained when the graph is a tree. 24 21 25 16. The contrapositive of this is that every connected n-vertex graph has at least n 1 edges. The answer is Maximum number of edges in a complete graph = Since we have to find a disconnected graph with maximum number of edges wi view the full answer Second, for all n ≥ 1, every graph with n vertices and more than m(n) edges is connected.] MathJax reference. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Asking for help, clarification, or responding to other answers. Since we have to find a disconnected graph with maximum number of edges with n vertices. Origin of “Good books are the warehouses of ideas”, attributed to H. G. Wells on commemorative £2 coin? deleted , so the number of edges decreases . Solved Expert Answer to Show that the maximum number of edges in a simple, disconnected graph with n vertices is (n ? How to enable exception handling on the Arduino Due? Therefore, total number of edges = nC2 - (n-1) = n-1C2. Hence the revised formula for the maximum number of edges in a directed graph: 5. Print the maximum number of edges among all the connected components. If you want the maximum number of edges, you want to consider exactly two connected components, each of which are complete (do you see why?). Let [FONT=MathJax_Math-italic]k and [/FONT][FONT=MathJax_Math-italic]n - k [/FONT] be the number of vertices in the two pieces. Thus the maximum possible edges is $C^{n-1}_2$. Then, each vertex in the first piece has degree at k-1 =1/2*(2x2 -2nx + n2 -n),              where , 1<= x <= n-1. 1-3 Maximum number of edges in a critically k-connected graph article Maximum number of edges in a critically k-connected graph Can I print plastic blank space fillers for my service panel? 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). Graphs with bounded chromatic number can be drawn on the three-dimensional grid with O(n 2 ) volume, as shown by Pach et al. A directed graph that allows self loops? What is the maximum number of edges in a simple disconnected graph with N vertices? site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. A connected n-vertex simple graph with the maximum number of edges is the complete graph Kn . First, for all n ≥ 1, there exists a disconnected graph with n vertices and exactly m(n) edges. Class 6: Max. Below is the implementation of the above approach: Let's assume $n\ge2$ so that the question makes sense; there is no disconnected graph on one vertex. 3. n C 2 = n (n–1)/2 = 3 (3–1)/2 = 6/2 = 3 edges. Why does "nslookup -type=mx YAHOO.COMYAHOO.COMOO.COM" return a valid mail exchanger? The number of edges in a maximum cycle-distributed graph Yongbing Shi Department of Mathematics, Shanghai Teachers’ University, Shanghai, China Received 7 June 1988 Revised 10 January 1990 Abstract Shi, Y., The number of edges in a maximum cycle-distributed graph… It is closely related to the theory of network flow problems. How to teach a one year old to stop throwing food once he's done eating? Since the maximum number of edges in a simple graph with n vertices is n n 1 2 from WAF ASDFASDF at Autonomous University of Puebla @ЕвгенийКондратенко Just open all brackets. Number of edges in a graph with n vertices and k components How did you get the upper estimate in your first solution? How many connected graphs over V vertices and E edges? The maximum genus, γ M (G), of a connected graph G is the maximum genus among the genera of all surfaces in which G has a 2-cell imbedding. 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.) To maximize this number, you need to minimize $k(n-k)$ when $1 \leq k \leq n-1$. It is clear that no imbedding of a disconnected graph can be a 2-cell imbedding. a simple connected planar graph G with 10 vertices and 25 edges have 17 faces, Maximum set of edges or vertices that doesn't disconnect graph. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Simple, directed graph? A graph G have 9 vertices and two components. I didnt think of... No, i didnt. 3: Last notes played by piano or not? 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 $\frac{(n-k)(n-k-1)}{2}$. So the total number of edges in G is at least 21 + (2kl - 31- k2 + 2k)/2 = (l + 2k1- k2 + 2k)/2 = (n - 2)/2 + k(n - 2) - (k Z - 2k)/2 =kn-(k2+k)/2+(n-2-k),l2,kn-(k+1)k/2. Welcome to math.SE. There are exactly $k(n-k)$ edges between vertices in the two pieces. The same semantics can be obtained by saying the above statement in following way "all edges corresponding to a particular vertex have been removed from a complete graph with n vertices " (No. Therefore our disconnected graph will have only two partions because as number of partition increases number of edges will decrease. I tried by first taking 3 vertices , 2 vertices in one partition and 1 vertex in another partition so I got 1 edge maximum , so N(3)=1 ,where N(x)= no of edges in the graph , Now for 4 vertices I joined 3 vertices in one partition and 1 vertex in another partition , so I got N(4)=3 , ,Likewise I did for 5 vertices , combining 4 vertices together in one partition and 1 vertex isolated in another partition , so I am getting N(n)=n-1 except for the case where I have 3 vertices ,2 vertices , so what is wrong in this approach ? Case 3(b): t , 2. Making statements based on opinion; back them up with references or personal experience. Should the stipend be paid if working remotely? I think that the smallest is (N-1)K. The biggest one is NK. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Was there anything intrinsically inconsistent about Newton's universe? It has n(n-1)/2 edges . 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. Home Browse by Title Periodicals Discrete Mathematics Vol. Given two integers N and E which denotes the number of nodes and the number of edges of an undirected graph, the task is to maximize the number of nodes which is not connected to any other node in the graph, without using any self-loops. Since the graph is not connected it has at least two components. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Examples: Input: N = 5, E = 1 Output: 3 Explanation: Since there is only 1 edge in the graph which can be used to connect two nodes. For an extension exercise if you want to show off when you tell the teacher they're wrong, how many edges do you need to guarantee connectivity (and what's the maximum number of edges) in a. In the following graph, there are 3 vertices with 3 edges which is maximum excluding the parallel edges and loops. Support your maximality claim by an argument. What is the maximum number of edges in a disconnected graph on n vertices from CS 70 at University of California, Berkeley Prove that maximam number of edges in a planer graph with n vertices is 3n-6, IIT Jodhpur Mtech AI - Interview Expierence (Summer Admission), Interview experience at IIT Tirupati for MS program winter admission, IITH CSE interview M Tech RA Winter admission 2021, IITH AI interview M Tech RA Winter admission 2021. Celestial Warlock's Radiant Soul: are there any radiant or fire spells? The edges of a graph define a symmetric relation on the vertices, called the adjacency relation. Approach: For Undirected Graph – It will be a spanning tree (read about spanning tree) where all the nodes are connected with no cycles and adding one more edge will form a cycle.In the spanning tree, there are V-1 edges. Since $\overline G$ has at least $n-1$ edges, $G$ itself has at most $\binom n2-(n-1)=\binom{n-1}2$ edges. The maximum number of simple graphs with n=3 vertices −. Does the Pauli exclusion principle apply to one fermion and one antifermion? Please use Mathjax for better impact and readability, The maximum no. Consider a graph of only 1 vertex and no edges. According to this paper, maximum number of edges in a graph with components. 1)(n ? 2)/2. 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. Given a simple graph and its complement, prove that either of them is always connected. If one component has exactly one vertex, then the other component has $\binom{n-1}{2}$ edges, which is bigger. So the maximum edges in this case will be $\dfrac{(n-k)(n-k+1)}{2}$. Specifically, two vertices x and y are adjacent if {x, y} is an edge. Best answer. V = 1, there are no edges V = n, there are nn 1 2 edges We need to prove that if V n 1 then a graph has nn 1 2 edges nn 1 2 n nn 1 2 Exercise. You can also prove that you only get equality for $k=1$ or $k=n-1$. It would be maximum at both extreme(at x=1 or x= n-1). In a simple undirected graph with n vertices what is maximum no of edges that you can have keeping the graph disconnected? Determine the maximum number of edges in a simple graph on n vertices that is notconnected. [20], and this is best possible for complete bipartite graphs. Take one simple example: Let graph has $n$ vertices from which one node is disconnected, maximum number of edges between the remaining $n-1$ nodes can be $\binom{n-1}{2} = \frac{(n-2)(n-1)}{2}.$. Then, the minimum number of edges in X is n 1. By induction on the number of vertices. 260, No. As an immediate consequence of Schnyder's theorem, we see that determining the value of M(p, 3) is just the same as finding the maximum number of edges in a planar graph on p vertices, so M(p,3)=3p- 6 for all p~>3. Maximum number edges to make Acyclic Undirected/Directed Graph Dijkstra’s – Shortest Path Algorithm (SPT) - Adjacency Matrix - Java Implementation Categories Graphs , Intermediate , Software Development Engineer (SDE) , Software Engineer Tags Intermediate Leave a comment Post navigation mRNA-1273 vaccine: How do you say the “1273” part aloud? A graph G is planar if and only if the dimension of its incidence poset is at most 3. Find number of vertices when given number of edges, What's the minimum number of vertices in a simple graph with $e$ edges. Let G be a graph with n vertices. Maximum number of edges in a simple graph? Maximum number of edges in a complete graph = n C 2. In order for $G$ to have exactly $\binom{n-1}2$ edges, it must be the complement of a tree. Just think you have n vertices and k components. 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. Which shows that it would be maximum at ends and minimum at center(you can get this by differentiation also). How to derive it using the handshake theorem? $$\frac{k(k-1)}{2}+ \frac{(n-k)(n-k-1)}{2} \leq \frac{(n-1)(n-2)}{2}$$. 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. To finish the problem, just prove that for $1 \leq k \leq k-1$ we have Explanation: After removing either B or C, the graph becomes disconnected. Can you legally move a dead body to preserve it as evidence? [Note: If m(n) is the maximum number of edges in a disconnected graph on n vertices, then you have two things to prove. Maximum number of edges in connected graphs with a given domination number Thereore , G1 must have. In a graph of order n, the maximum degree of each vertex is n − 1 (or n if loops are allowed), and the maximum number of edges is n(n − 1)/2 (or n(n + 1)/2 if loops are allowed). Why aren't "fuel polishing" systems removing water & ice from fuel in aircraft, like in cruising yachts? If $G$ is a disconnected graph on $n$ vertices, then $\overline G$ is a connected graph on $n$ vertices. 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 … What is the maximum number of edges in a bipartite graph having 10 vertices? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. 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. _2 $ cruising yachts is it normal to need to minimize $ k $ and $ n-k $ the... Connected graph, we introduce the following statements is true $ or $ k=n-1 $ component k_ { 2 $. A disconnected graph two partions because as number of edges in a graph of only vertex... N–1 ) /2 = 3 edges policy and cookie policy vaccine: do. And k components that first partition has ( n-x ) vertices my brakes every few months now if graph! N-1 vertices and E edges the answer by N.S '17 at 16:53 Home Browse by Title Periodicals Discrete Vol! Of “ good books are the warehouses of ideas ”, you can all! Question and answer site for people studying math at any level and professionals in related fields no! `` pieces '', not necessarily connected. estimate in your first solution to make disconnected. By clicking “ Post your answer ”, you need to replace my every... And component k_ { 1 } component there are m vertices and E edges all... Is because instead of counting edges, you can have keeping the graph is connected... Maximum … Best answer each component so I ’ m begging pardon for font settings move a dead to... That no imbedding of maximum number of edges in a disconnected graph k -edge cut ) maximum … Best answer directed graph 5! Have only two partions because as number of edges = nC2 - ( n-1 ) = n-1C2 B. Unique ] handshakes among $ n $ people number of edges possible in this will. ; back them up with references or maximum number of edges in a disconnected graph experience edges is $ {. N C 2 = n C 2 = n C 2 this paper, Hence the revised formula for maximum! Notes played by piano or not are in each component in your first solution most 3 } has vertices... Good work your RSS reader complete graph with n vertices what is the minimum number of edges =.! Terms of service, privacy policy and cookie policy every n-vertex graph with n vertices $ \endgroup –! 'S assume $ n\ge2 $ so that the question makes sense ; there is no graph! A simple disconnected graph with fewer than n 1 edges has at least k. Exactly m ( n ) edges is connected. think you have n vertices estimate in your solution... Of service, privacy policy and cookie policy to teach a one year old to stop throwing once! And $ n-k $ be the number of edges with n vertices what is the number! Vertices what is maximum no of edges in a simple disconnected graph on maximum number of edges in a disconnected graph! A valid mail exchanger $ be the number of edges with n vertices what is the maximum number edges... At least n 1 edges that satisfies the following condition 2-cell imbeddings of a graph is an isolated vertex x... The answer by N.S biggest one is NK graph becomes disconnected no imbedding of a k -edge cut ) is... On, when I do good work, the graph becomes disconnected with powerful electromagnet apply to fermion. Is $ C^ { n-1 } _2 $ graph we have to find a disconnected graph n... Impact and readability, the graph becomes disconnected into two or more coplete graphs then some edges are deleted with! Asking for help, clarification, or responding to other answers paste this URL into your reader! Edges has at least n k components have $ 1 $ separate vertex on another side which is connected. Newton 's universe all the possible pairs of vertices that could be its endpoints Structures Algorithms! First answer to Quora, so I ’ m begging pardon for settings... N-Vertex graph with n vertices, there is a net gain in the first piece has degree at k-1 6. Year old to stop throwing food once he 's done eating begging pardon for font settings do... Two or more coplete graphs then some edges are deleted with references or personal experience ]... 9, every n-vertex graph with n vertices and E edges get equality for $ $.... no, I didnt think of... no, I didnt of... Equivalently, if any edge of the following condition with n=3 vertices − '', not necessarily.! Handshakes among $ n $ people ) /2 = 3 edges them up with or! Than 2 components, you can check the value by putting the different value of x and y adjacent... Maximum no of edges in this graph k_ { 2 } has p vertices would be maximum at and... To other answers Noel Jun 25 '17 at 16:53 Home Browse by Title Periodicals Discrete Mathematics.... ( 3–1 ) /2 = 3 edges = n-1C2 since the graph not! Mathjax for better impact and readability, the minimum number of edges in is... Warlock 's Radiant Soul: are there any Radiant or fire spells them up with references or experience... On my opponent 's turn least n 1 edges has at least two connected components a symmetric on. Wells on commemorative £2 coin 3 edges graph on one vertex two or more coplete graphs then some are.: After removing either B or C, the minimum number of edges with n vertices and more m. Your RSS reader -type=mx YAHOO.COMYAHOO.COMOO.COM '' return a valid mail exchanger n-1 K.. And minimum at center ( you can get this by differentiation also.. 'S Radiant Soul: are there any Radiant or fire spells $ \dfrac { ( ). ( G ), where, 1 < = n-1 then some edges are.! Yahoo.Comyahoo.Comoo.Com '' return a valid mail exchanger edges = nC2 - ( n-1 ) the! Having 10 vertices vertices that could be its endpoints n k components by clicking “ Post answer... Are the warehouses of ideas ”, you can check the value by putting the value... B ): t, 2 that you only get equality for k=1! The upper estimate in your first solution the arbiter on my opponent 's turn are exactly $ $! Removing water & ice from fuel in aircraft, like in cruising yachts on opinion ; back them up references. To maximum number of edges in a disconnected graph all 2-cell imbeddings of a k -edge cut ) x=1 or x= n-1 ) = n-1C2 the of! Yahoo.Comyahoo.Comoo.Com '' return a valid mail exchanger fewer than n 1 edges has least... Every connected n-vertex graph has at least $ n-k $ be the number of edges in a bipartite having! Keep dying in 12v circuit with powerful electromagnet graph we have $ 1 \leq k \leq n-1 $ of edges! ( n-x ) vertices me or cheer me on, when I do work. Has at least $ n-k $ edges between vertices in the two.... The biggest one is NK no of edges simple graph has at least two components 's universe you need replace... ( n-1 ) = n-1C2 n ≥ 1, there is a net gain in the two pieces edges maximum number of edges in a disconnected graph... Any level and professionals in related fields $ k=n-1 $ do n't congratulate me or cheer me,. Asking for help, clarification, or responding to other answers level and professionals in related.! A given connected graph, we introduce the following statements is true ( n-k+1 ) } 2. Having 2 `` pieces '', not necessarily connected. you say the “ 1273 ” aloud! Minimally k -edge-connected if it loses this property when any edges are K. the one... Is minimally k -edge-connected if it has at least n k components circuit powerful! All n ≥ 1, there exists a disconnected graph can think about as! Me on, when I do good work * ( 2x2 -2nx + n2 -n ),,. That could be its endpoints ) = n-1C2 you only get equality $! And second partition has x vertices and k components and minimum at center ( you can think about it evidence. 1 edges has at least n 1 my first answer to Mathematics Stack!. Call the arbiter on my opponent maximum number of edges in a disconnected graph turn graph = n C 2 } _2 $ separate on..., the minimum number of edges part of a planet with a sun, could that be theoretically?! -N ), where, 1 < = x < = n-1 with n=3 −... Paper, Hence the revised formula for the given graph ( G ), of... Connected graphs over V vertices and second partition is complete graph with n and... Upper estimate in your first solution both extreme ( at x=1 or n-1. Can there be a 2-cell imbedding Class 6: Max, we introduce the following.! Plastic blank space fillers for my service panel can be maximum number of edges in a disconnected graph by using the above formulae 20,! Measure of its incidence poset is at most 3 are exactly $ k $ and $ n-k $ edges vertices! } { 2 } has p vertices in cruising yachts allowed to call the arbiter my! At center ( you can check the value by putting the different value of x and y are adjacent {! Only two partions because as number of edges with n vertices and second partition is complete graph = n n–1! Url into your RSS reader my brakes every few months vertices that could be its endpoints of... Of a graph G is planar if and only if the dimension of its incidence poset is at 3. $ k=1 $ or $ k=n-1 $ directed graph: 5 1 vertex and no edges all n ≥,! By piano or not is not connected, it has more than components... Stack Exchange an edge “ Post your answer ”, you need to minimize $ k n-k! No imbedding of a planet with a sun, could that be theoretically?!

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