Graph Definition and 999 Threads

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection.
Plane graphs can be encoded by combinatorial maps or rotation systems.
An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a planar map. Although a plane graph has an external or unbounded face, none of the faces of a planar map has a particular status.
Planar graphs generalize to graphs drawable on a surface of a given genus. In this terminology, planar graphs have genus 0, since the plane (and the sphere) are surfaces of genus 0. See "graph embedding" for other related topics.

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1. Mathematica - How to Graph Direction Fields of ODEs?

So, I'm working on ODE at this point, and unfortunately I don't have the time to learn all of the Mathematica input for at least a few months. But, I was hoping someone could tell me how to graph direction fields of ODEs, either with the Mathematica input or free-form? Looking through the...
2. lim x-->0 ##\frac{x tan2x -2xtanx} {(1-cos2x)^2}##

I simplified this function to ##\frac{1}{2} (\frac{x tan^3(x)} {(sin²x)²(1-tan²x)}## Now further can I not write ##1-tan²x## as ##\frac{cos2x} {sin²x}## ? If I do that I get ##\frac{1}{2} (\frac{x tan^3(x)} {sin²x cos2x}## On graphing this on desmos I get two different graphs for these...
3. I Correlation vs causality implied by a graph

The graph below shows a very clear, but not perfect, correlation between the red line and the blue line. That fact says nothing whatsoever about causality. BUT ... if you add in the additional fact that there is absolutely zero chance that the blue events cause the red events and a quite...
4. Slope of N - t graph of radioactive decay

I am not really sure how to interpret the slope. The equation is: $$N=N_o e^{-\lambda t}$$ If the graph is N against t, then what is the slope? I can find the slope if the graph is log: $$log N=log N_o -\lambda t$$ So if the graph is log N against t, then the slope is ##-\lambda## But if...
5. Finding focal length from di v. M graph

I tried di= f +fM, but that would mean slope and y int is the same when it’s not.
6. Which software to use to create this graph?

So I have a dataset with 3 variables -Time -Quantity 1 -Quantity 2 Currently I have all the data in an excel sheet. So for example, I can easily in excel create a line graph that shows me quantity 1 on the Y axis and time on the X axis... however I want to see all 3 at the same time. So my...
7. Half life of multi chain decays

I am not really too sure where to start with this one, I would guess that the half life of the graph is very similar to the half life of the X to Y step, but at the same time I am unsure of how I would prove it. Any tips or reading material? Thanks!
8. Vertical Velocity Graph versus Time on a Slope

I understand that through process of elimination the only plausible solution is (E), but a question that rises up: When the ball bounces, does the velocity change from negative to positive instantly (as shown by the dotted lines) or gradually (a very small time period, but still solid line)?
9. Background Analysis for MCBEND Simulation

I have compared my simulation results with the experiments I conducted in the laboratory. Here I want to add a background for the graph I obtained from the simulation. Can I do it? I would be very happy if you could help me. I used MCBEND as the simulation program.
10. Probability related to cumulative distribution function

I have tried to answer all the questions but I am not that sure with my answer. That's the graph of ##F_X (x)## (I think) (i) P (X ≤ i) = ##\frac{i^2}{N^2}## and P(X < i) = 0 All of these are based on the graph (ii) P(X = i) = P(X ≤ i) - P(X < i) = ##\frac{i^2}{N^2}## Are my answers...
11. About critical damping resistance of a ballistic galvanometer

At critical condition, ##\omega=0## so time period will be infinite and so will be ##\lambda##.Therefore, the critical resistance will be the corresponding resistance(plus galvanometer resistance)of the asymptote of ##\lambda## vs. ##R_2## graph(the graph is a rectangular hyperbola). But here's...
12. I Find Smallest Subtree of G w/ Nodes from S

I have a unique problem and looking for a known fast algorithm for it. I have an unweighted but directed graph G. I then have a subset collection of nodes S existing in G. I want to find the minimum sub tree in G such that it contains all the nodes in S. I so far found Chu-Liu/Edmonds...
13. Analyse motion of an oscillator: x(t)=0.2cos(12*pi*t)

Hello. I have tried to solve it using x-t Graph. We know that period of this function is ##T=\frac {1}{6}s##. Then I've used ##x(t)=0## to find the times in which the oscillator is at ##x=0##: ##t=\frac {k}{12} + \frac {1}{24}## for ## k \in Z.## Now I can draw x-t graph. We should check time...
14. Radii of stacked circles inside the graph of y = |x|^1.5

(a) The hint from question is to used geometrical argument. From the graph, I can see ##r_1+r_2=c_2-c_1## but I doubt it will be usefule since the limit is ##\frac{r_2}{r_1} \rightarrow 1##, not in term of ##c##. I also tried to calculate the limit directly (not using geometrical argument at...

16. M

Graph of f' using graphical methods

For this Problem 5, The solution is, However, I though the graph of f' would have end behavior more like, Does someone please know whether I am correct? Many thanks!
17. Calculating electric charge from graph (capacitor)

Apparently, we need to integrate the functions from 0 to the time when it is fully charged. However, I integrated in terms of t so the soultion (according to a graph programme) should be around 236 Vs but I don’t see how this could help me.
18. Combination of 2 formulas for a graph

i need to combine the two to make a new 3rd one I think the t element is the X axis scale ? not quite sure how to do this, suggestions please ?
19. I Relationship between the Different Frequency vs Decibel Graph Peaks

In the popular answer for the coin-mass question of Physics Stack Exchange, I am wondering what are the correlation between the first red peak at around 9kHz and the second red peak at 16kHz. I first thought that they are consecutive harmonics but there was no way of proving it as I do not know...
20. M

Points of inflection on acceleration graph

EDIT: For this part(b) of this problem, The solution is However, isn't there more points of inflection than just ##t = 3,5 s ##? Points of inflection is when ##x'' = a = 0## so it should be ## 3 ≤ t ≤ 5 s## I also have a question about part(d): The solution is However, could I tried...
21. M

Graph for sinusoidal wave travelling to the left

For part(a) of this problem, The solution is, I don't understand why they assume on the graph where that the waveform is during it's phase. For example, could it not also be correctly drawn as shown in red: Could it not? Many thanks!
22. M

Discontinuous voltage graph for bullet going through metal coils

For this problem, The solution is, However, dose anybody please know why the graphs for ##V_1## and ##V_2## are discontinuous where they cross the time axis? Many thanks!
23. M

Drawing the Graph: Analyzing Solutions and Questions

For this problem, , The solutions is, However, why did they not draw vertical lines for the graph like this: ? Thank you!
24. Formula for Xo and Yo for graph of quadratic function

Can someone please tell me where I am wrong. I am learning how to write ##a^{2} + bx + c## in this form ##f(x)= a(X-X_0)^{2} +Y_0##. The method used in my textbook is a reduction to the perfect square. And it goes like this: ##f(x)=ax^2+bx+c## ##=a[x^2+\frac{b}{a}x]+c## ##=a\left [...
25. I Triple equation for integral on a graph

Hi, so I'm trying to find the volume of a shape using integral, I found the equation of one plane in 3D space but the second one is something like that, which I cannot write in integral as a function: ##\frac{2(2x-a)}{a}=-\frac{2(6y-a\sqrt3)}{a\sqrt3}=\frac{2z-a\sqrt3}{a\sqrt3}## In the 3D...
26. Finding acceleration from Velocity vs Position graph

The answer is E. I was initially very confused as to why the answer was not A but realized that the graph was velocity vs position (rather than velocity vs time) which means I can't simply take the derivative of the given graph. One thing I tried was writing out the equation first(c being a...
27. Analyzing the Effects of Battery Position on ##C_m-\alpha## Graph

I am trying to determine how the addition of a counterweight affects the ##C_m-\alpha## (in longitudinal direction) graph for a model plane project, where the counterweight can be considered as the battery. Suppose we try to place the battery at the back of the current CoM. The new CoM will...
28. Estimating the Gradient of a Graph: Student's Attempt

Ok this is a question that i am currently marking...the sketch is here; In my mark scheme i have points ##(1,2)## and ##(3,5)## which can be easily picked from the graph to realize an estimate of ##m=1.5## where ##m## is the gradient ...of course i have given a range i.e ##1.6≥m≥1.2## Now to...
29. Functional representation of the oscillating graph

Hi; This is in fact not a homework question, but it rather comes out of personal curiosity. If you look at the graph of the two functions in the image attached, what is the simplest functional representation for such a symmetrical pattern?
30. A What does Rovelli mean with "oriented and ordered graph"?

Hi Pfs Rovelli writes this in his book (Qunatum Gravity) about spin networks: Given an oriented and ordered graph there is a finite disgrete group of maps that change its order or orientation and that can be obtained as a diffeomorphism. A link is equipped the source and target functions. this...
31. How does energy affect the melting point of water?

Im not really sure how to answer this question. Would you have any ideas?
32. What does the 'power' vs 'time' graph look like for a 3 phase AC supply?

Hi, For years I wanted to understand why we have a 3 phase supply and not a 2 phase supply in AC. In Quora I found an interesting answer and was convinced about the purpose mentioned: Answer to Why there is no two phase electrical inputs instead of three phase and single phase? by Paul...
33. B What Math Class Teaches us how to Draw a Graph?

What math the drawing a graph, for example: y = x^2, is taught? Is it algebra, calculus, or any other else? Thanks
34. Graph Reading: Get it Right the First Time!

I'm fairly certain that the first one is marked correctly. Should the second one be where there are red marks or is it correct as it was marked previously? Any advice with graph reading?
35. Is the Answer Sheet Wrong? A Closer Look at Velocity-Time Graphs

So I am supposed to look at the graph at part C and determine the total distance travelled. This is the answer sheet of the textbook. But I feel like the textbook is wrong here. But then I needed a confirmation if others can agree that it is wrong or not cause now I am very confused. My answer...
36. Graphical Impulse calculation from the graph of force verus time

hello i would like to get some help with this problem. At first it try to calculate the impulse by the area but i found it too difficult Then i try to solve it by the forumla J= F(t-t0), but the problem is that i don't know what F is so i try to solve it like this F10) = 8 F(33) = -13 so EF =...
37. B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?

Hi, PF For example, ##\sin{x}=O(x)## as ##x\rightarrow{0}## because ##|\sin{x}|\leq{|x|}## near 0. This fits textbook definition; easy, I think. But, Taylor's Theorem says that if ##f^{(n+1)}(t)## exists on an interval containing ##a## and ##x##, and if ##P_{n}## is the ##n##th-order Taylor...
38. I I want to know which software was used to create vector calculus graph

Hi. I have the Marsden an Tromba vector calculus book 6th edition. I was wondering which software was used to create the books graphs. I attach two graphs as an example. Thanks
39. B Why is Delta y not equal to dy for infinitesimally dx on the graph?

Hello! As is known, \Delta y = dy for infinitesimally small dx. It's true. But if we have graph we may see that \Delta y isn't equal to dy even for infinitesimally small dx. Why is that so? Thanks!
40. Find the set of possible values of ##a## - Modulus Graph

My interest is on question 9. b(i) Find the question and solution here; I understand that ##a## should be less than ##2## because when ##a=2##, the two equations shall have same gradients which implies that the two lines are parrallel to each other. Now to my question, this solution does not...
41. B What is this part of a graph on the axis called?

On this graph on the Y-axis just above the origin is a "zigzag" mark (highlighted with the red circle) which represents a discontinuity as the value of Y jumps from 0 to 130. Does this mark have a formal name?
42. How to calculate distance traveled if you know a(v) graph equation?

I know that you can find the distance traveled by integrating v(t), but I can't find a way to convert a(v) into it. I tried derivating and integrating the a(v) equation, but I don't know what the results mean.
43. Visualizing Series Bandpass Filters: A Graphical Approach

this is the circuit this is the theoretical graph
44. B Just for fun: Cubic Graph Plot

Desmos.com is a great online graphing utility which I'm sure is familiar to many PF users. I wanted to experiment with the Newton-Raphson method using it so chose solution of cubic graphs as an example. The graph shows a variable cubic on which all turning points and intercepts are calculated...
45. Comp Sci Creating a City Graph with Nodes & Edges

Edit: Trying to fix the formatting... class Node(object): def __init__(self, name): self.name = str(name) def get_name(self): return self.name def __str__(self): return self.name class Edge(object): def __init__(self, src, dest): self.src =...
46. Determining the Yield strength and Ultimate tensile strength of HDPE

Help would very appreciated. Also I am new to this website so if something seems wrong please tell me :)
47. Making a Velocity vs Time graph for a bullet

Here is my working in case it helps :)Ignore the v = 0.6/0.0001 = 6000m/s
48. Question regarding a decay spectrum graph

I am not sure what is the description of Channel.
49. Spectral Graph Clustering: where does the 'scoring' function come from

Hi, I was reading through some slides about graph clustering. In the slides was a very short discussion about 'eigenvectors and segmentation'. I don't quite understand where one of the formulae comes from. Context: We have some undirected graph with an affinity matrix (i.e. weighted adjacency...