Yes. Tensor product is another name for Direct product.
@kevon217Ай бұрын
Great playlist so far. Lot to keep track of, but really appreciate the attention to detail and visual examples/comparisons.
@VitalSineАй бұрын
Thank you, I'm glad to hear you like the playlist!
@kevon217Ай бұрын
Clutch video. Thanks!
@adaogmailcomАй бұрын
Vital, do you by any chance happen to have a Microsoft Excel Spreadsheet that can calculate this Sterling numbers of the second king? If you do, I'd appreciate if you can share it with us. Thanks.
@tarekbenslimane4620Ай бұрын
Nice videos ! Congrats ! Couldn't help but subscribe !
@queenlowkey94812 ай бұрын
I hate this whole combinatorics topic🚮
@jaja-qt4gm3 ай бұрын
great video
@chaithanyachaithanya-qc2fu4 ай бұрын
In fractional coloring,when we apply fractional colors to each of the vertices of the graph we can't get any fractional value of chromatic number,why?
@thecyberlocal4 ай бұрын
Your videos are fascinating. I wanted to solve the exercise but I'm not smart enough yet. I would LOVE it if you made a prep course in graph theory to teach it to people who haven't taken college, so they are ready for your videos. Regardless, you're awesome!
@VitalSine4 ай бұрын
Thank you so much for the kind words! It's a great idea to make a prep course, I haven't been able to make any new videos recently, but I'll definitely make more preparatory content in the near future. I also wanted to let you know I have a couple of introductory graph theory videos on the channel in case you haven't studied introductory graph theory and have not found them yet, but there is definitely more to be done along the lines of prerequisite content. Also, I'm not sure what level of mathematics you are studying right now, but if you're interested, a great textbook for all the prerequisites of graph theory would be "Discrete Mathematics" by Susanna Epp, it's a very good introduction to logic, sets, functions, combinatorics, (a whole bunch of other concepts too) and also has a section introducing graph theory. If you want to get into the math that graph operations, such as graph products, are built from, see the book "A Book of Abstract Algebra" by Pinter, that book is fantastic and if you like graph products you will quickly see the relevance to graph theory while reading it.
@thecyberlocal4 ай бұрын
Here are my notes. Hope they help someone. Injective (one-to-one) Function: Maps domain elements to distinct codomain elements. Surjective (onto) Function: Maps all elements of a domain to all elements of a codomain. Bijective (one-to-one correspondence) Function: Both injective and surjective.
@thecyberlocal4 ай бұрын
This video really helped. I look forward to watching the rest.
@TwentyNineJP4 ай бұрын
I just came across the concept of hypergraphs yesterday, and I think they're exactly the solution I've been looking for for project I've been stuck on
@VitalSine4 ай бұрын
Awesome! I'm curious, what kind of project are you working on if you are able to share?
@TwentyNineJP4 ай бұрын
@@VitalSine KZbin filtered my reply because I linked to the site, which is frustrating 😅 It's a tool that evaluates the output function of simplified digital integrated circuit topologies ("stick diagrams"), primarily intended for students. Through a lot of trial and error, the internal model has converged on something that is essentially a directed hypergraph, but the evaluation algorithm (stepping through vertices and determining their logic levels) is very complex and has a deeply rooted bug that sometimes manifests when VDD (logic high) and GND (logic low) are shorted together. I think that applying existing hypergraph theory might help me greatly simplify and debug the algorithm. Since I can't post a link, I'll describe the URL. The domain name is "stixu", the TLD is "io". If you go to "/test", it'll run the testbench so you can click on the individual test cases to see example circuits. Most of the test cases have 8 results - that's for all rotations and reflections of the circuit. The failing tests (aside from the intentionally failed test case intended to make sure the testbench itself works) are rotation and reflection dependent
@TwentyNineJP4 ай бұрын
@@VitalSine KZbin filtered my reply *twice*, which is frustrating 😅 I'll try again without referencing how to get to the project It's a tool that evaluates the output function of simplified digital integrated circuit topologies ("stick diagrams"), primarily intended for students. Through a lot of trial and error, the internal model has converged on something that is essentially a directed hypergraph, but the evaluation algorithm (stepping through vertices and determining their logic levels) is very complex and has a deeply rooted bug that sometimes manifests when VDD (logic high) and GND (logic low) are shorted together. I think that applying existing hypergraph theory might help me greatly simplify and debug the algorithm.
@VitalSine4 ай бұрын
@@TwentyNineJP Fascinating, best of luck with your project!
@ProbablyKen4 ай бұрын
Good video, thanks! Mistake at 4:30. Arrow from (1 3 2) should point to (2 1 3) instead of (3 1 2).
@NaveenaBM-bj9ur4 ай бұрын
Consider the graph with two vertices joined by an edge, vertex A is colored with red and blue, vertex B is colored with green,yellow and pink is it possible in fractional coloring,if it possible then what is the b-fold chromatic number?
@VitalSine4 ай бұрын
It would be possible for a fractional coloring where we allow for sets of a different number of colors to be used on each vertex, but it would not be under the category of a b-fold coloring. The b-fold chromatic number of the complete graph on 2 vertices would be 2*b.
@indusuresh70976 ай бұрын
Hi, thank you for making this video which is very useful in my studies. But i have one doubt in line graph that is what if the intersection of hyperedges contain more than one vertices? for example let e_1 and e_2 are hyperedges and there intersection is more than 1 vertices, so then what about the line graph?
@VitalSine6 ай бұрын
Hi, great question. If the intersection is more than 1 vertex, in the line graph e_1 and e_2 are still made adjacent. The definition of adjacency in a line graph only requires that the edges share at least one incident vertex.
@inscitia6 ай бұрын
"Underlying graph of an Abstract Simplicial Complex hypergraph is the hypergraph's 2-section." It seems that graph associations (transformations), like this one between a graph and its simplex graph, are naturally described using hypergraphs. Graph powers is other case: resulting edges are encoding a subset of vertices contained in the path with size equal to the exponent of the power graph.
@quan21787 ай бұрын
Thanks. it helps me a lot
@NamNguyen-pq7sp7 ай бұрын
you missed the parentheses at g^j(x)^mj at 1:39 in the video
@KanalmitHut7 ай бұрын
Hello Vital Sine, I appreciate you creating Videos on Topics of Graph Theory that otherwise don’t get much attention on KZbin. Out of interest I looked into the article by Махнев from Ural you linked on arxiv. I found a paper from Faber and Keegan called "The existence of a moore graph of degree 57 is still open", this paper is a direct response to the linked and should disprove the extend of its findings. Do you know any further developments on this matter? Regards KmH
@jj-wp8dt7 ай бұрын
Such an amazing explanation. Thank you
@emanabdelhaleem75617 ай бұрын
thank you
@user-qq6do5gy8i7 ай бұрын
Your videos are informative and It'll be a great help if I get the presentation as pdf.
@VitalSine7 ай бұрын
Sure thing, here it is: drive.google.com/file/d/1MBhqmHa6rgg1q0xqze4llHf_PXhNLSo1/view?usp=sharing
@gerardsagliocca62928 ай бұрын
Going to fast.
@gerardsagliocca62928 ай бұрын
You sound like you are in a Chamber box.
@soup_is_good_food57438 ай бұрын
Great job! Very clear and easy to follow, yet covering the essential points precisely. Thank you, I appreciate your work.
@Channel-nu1bv9 ай бұрын
I enjoyed this video, and subscribed.
@Channel-nu1bv9 ай бұрын
Mindblowing! Ultra clear and helpful! Thanks a lot!
@yusufturhan83679 ай бұрын
very interesting topic and great explanation!
@khaledfada94019 ай бұрын
on minute 4:39 there is a typo when it says (D,G,E) is a cycle, you should add D after the E. Thank you.
@samiyaramzan63039 ай бұрын
What is difference between strong product , lexicographic product and cartesian product in graph??
@VitalSine9 ай бұрын
Great question. The difference is in the adjacency rules. All three products take two graphs, G and H, as input, and all three products output a graph whose vertex set is the set cartesian product of the vertex sets of G and H. In the graph cartesian product, the vertices (u, v) and (u', v') are adjacent if either u = u' and v is adjacent to v', OR u is adjacent to u' and v = v'. That is the adjacency rule for cartesian products. In the strong product, the vertices (u, v) and (u', v') are adjacent if either u = u' and v is adjacent to v', OR u is adjacent to u' and v = v', OR u is adjacent to u' and v is adjacent to v'. It is the same as cartesian product but with one extra condition (the last one). In the lexicographic product, the vertices (u, v) and (u', v') are adjacent if either u = u' and v is adjacent to v', OR if u is adjacent to u' (regardless of the relationship between v and v'). The lexicographic product will generally be the most dense of the three. Hope this helps!
@pswjt12669 ай бұрын
Hi, this series is very useful thank you for making this! I have a very specific technical question if you don't mind me asking. I work on an algorithm which analyzes the 3-line graph of our hypergraph structured data, which is the line graph except you only include edges for hyperedge intersections which have at least 3 vertices. I have found that if you associate every edge in the 3-line graph with the vertices that form the corresponding connection between hyperedges, you can define each set of these vertices as a hyperedge and construct another hypergraph, which seems to have very interesting properties. Does that make sense? Is there some hypergraph theory that describes such an operation? I have not been able to find it thus far. Thank you again!
@pswjt12669 ай бұрын
@@VitalSine This does indeed seem close to what I have in mind! Thank you so much!
@timelygoose9 ай бұрын
Thank you sir, such a great concise explanation
@VitalSine9 ай бұрын
Most welcome!
@wangealork476810 ай бұрын
I'm an undergraduate student and will go for a PhD in combinatorics, this video really helps. Thanks a lot!
@RadhaKrishnan-ru8py10 ай бұрын
This videos help me to understand very easly this concept 🎉🎉
@RadhaKrishnan-ru8py10 ай бұрын
Thanqq.... So much sir your attempt help me to complete my project successfuly.... Thanks alode❤❤
@ZhixuanLuo-hl9fk10 ай бұрын
Thanks a lot!!! Finally grasp this concept!
@juanzuluaga338811 ай бұрын
Video is not clear. It jumps to formulas that imply definition of shortest paths, without providing explanation by visualization. "Percolation" itself is a physical process ("the process of a liquid, gas, etc. moving gradually through a surface that has very small holes or spaces in it") that could have been visualized directly.
@golden_smaug11 ай бұрын
Books' definitions are usually quite cumbersome but thanks to this video I got a good grasp of what it is a cartesian product, thanks! :)
@MyMapTV11 ай бұрын
helped me very much, great explanation! thanks alot
@VitalSine11 ай бұрын
Glad it helped!
@kipling1957 Жыл бұрын
Fell at the first post. Can’t you just describe something in simple language for the non-mathematician? Vertices? Edges?
@darklordvadermort Жыл бұрын
any interest in hierarchical navigable small worlds (probabistic/best in class nearest neighbor search algo) or tangential topics?
@VitalSine Жыл бұрын
Hello, I'm not familiar with that concept, sorry. However, it does look interesting from what I've read since searching for it online!
@pierrelagrange9104 Жыл бұрын
Very cool video :) I do share @darklordvadermort intuition concerning the relevance of percolation measures in HNSW (as a tool for optimisation of vector database). Indeed, the more "percolative" is a node in a knowledge graph, the more it will have an impact on the efficiency of query, and the more it will help to find a nearest neighbor. Could be a very interesting topic though !
@navneetsinha451 Жыл бұрын
I want to create a hypergraph in Python which contains 100s of nodes, and user defined edges. Is it possible to do this ?
@VitalSine Жыл бұрын
This is certainly possible, I would suggest you look into HyperNetX for an existing Python library for working with hypergraphs, pnnl.github.io/HyperNetX/index.html
@navneetsinha451 Жыл бұрын
@@VitalSine Thanks for your reply. If I want to draw a hypergraph having hundreds of nodes then I can use Matrices may be, however it will be a sparse matrix. Also space consuming code!
@navneetsinha451 Жыл бұрын
@@VitalSine Can you suggest any other Data structure I can use. That would be of great help!!
@VitalSine Жыл бұрын
You could use a list for the nodes, you can use a dictionary to represent your collection of edges, such as {"e1" : [1 , 2, 3], "e2" : [2, 3]}.
@tanchienhao Жыл бұрын
Cool and informative video!
@azizautop995 Жыл бұрын
Great video and explanation.
@mymathware Жыл бұрын
Please do you have any video tutorial on cluster hypergraphs?
@VitalSine Жыл бұрын
I do not have any videos on cluster hypergraphs yet, but I will see if I can make some new videos about them 👍
@HarpreetSingh-ke2zk Жыл бұрын
Good explanation. However, as n increases, the manual way to look for integer partitions becomes more difficult. How can one apply this formula to a larger n?
@VitalSine Жыл бұрын
Great question! For larger n, you could use an algorithm to generate a list of integer partitions, convert to the m_i and step through the list, plugging each element into the formula and summing. This link has some information on creating a list of integer partitions: stackoverflow.com/questions/400794/generating-the-partitions-of-a-number
@mymathware Жыл бұрын
Thank you so much, I have learnt so much on hypergraphs since I started watching your videos. I have an observation, in this first example, is the vertex v4 not incident to edge e1 and e2? Because you didn't connect v4 to e1 in the bipartite graph. Or is there a rule that disallows this?
@VitalSine Жыл бұрын
I'm happy to hear you enjoyed the hypergraph videos. Good catch, that was a mistake on my part, v_4 should connect to e_1 in the bipartite graph.
@ansumanc Жыл бұрын
I wish i discovered this channel earlier, nevertheless, I'm gonna go through your previous videos.
@B312XC Жыл бұрын
I realized that you don't go to university to learn. You go there to cram as much knowledge as possible in 4 years and the actual learning comes from people online who are far better at teaching.