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. N

    Use Graph To Investigate Limit

    Summary:: Use Graph To Investigate Limit Use the graph to investigate the limit of f(x) as x tends to 0. Let me see. I got to use the graph to investigate the limit of f(x) as x tends to 0 from the left and right. Let y = f(x). The given function can also be expressed as f(x) = | x |. The...
  2. karush

    MHB What is the graph of y = sin^2 x + cos^2 x?

    ok image to make sure I didn't have typo's but I am clueless... I thot $y=\sin^2 x+cos^2 x$ was the graph of $y=1$
  3. brotherbobby

    The integral of a function ##f(x)## from its graph

    Problem statement : I start by putting the graph of (the integrand) ##f(x)## as was given in the problem. Given the function ##g(x) = \int f(x) dx##. Attempt : I argue for or against each statement by putting it down first in blue and my answer in red. ##g(x)## is always positive : The exact...
  4. brotherbobby

    The first and second derivatives at various points on a drawn graph

    Problem statement : The function ##y = f(x)## is given above. Question 1 : Locate the points at which the ##\text{first derivative}## of ##y## with respect to ##x## is ##\text{non-zero}##.##\\[5pt]## At points of extrema, like A, C and D, the derivative is zero. Hence the derivative is non...
  5. uranium92

    A Graph analysis problem to find when the acceleration is zero

    I had deduced that B,E,H are the places where acceleration will be zero, but when I read the solutions it showed that K also has a = 0. It said it had maximum slope and then said a = 0. But I couldn't understand why? Please help.
  6. P

    Understanding the Forces Behind Part B of the Graph

    I am only asking about the answer to part B, but reading through part A may give some some context/familiarity. Below is the answer to part B: I largely understand the graph except for 1 part. My understanding is as such: At first, ##x = \frac {\mu_k m g } {k}##. Force exerted by the...
  7. M

    MHB Uniform convergence - Length of graph

    Hey! :giggle: We define the sequence of functions $f_n:[0,1]\rightarrow \mathbb{R}$ by $$f_{n+1}(x)=\begin{cases}0 & \text{ if } x\in \left[ 0, \frac{1}{2n+3}\right ]\\ |2(n+1)x-1| & \text{ if } x\in \left [\frac{1}{2n+3}, \frac{1}{2n+1}\right ] \\ f_n(x) & \text{ if } x\in \left...
  8. N

    What is the intersection point of two objects on a position vs time graph?

    The graph: 1) $$v_1 = \frac{\Delta{x}}{t} = \frac{5 - 3}{3} = \frac{2}{3}$$ $$2 = \frac{\Delta{x}}{t} = \frac{4 - 0}{5 - 1} = 1$$ $$\frac{v_2}{v_1} = \frac{1}{2/3} = \frac{3}{2} = 1.5$$ 2)Points of intersection of the lines with the x-axis: ##I## (0; 3) and ##II## (0; -1), thus $$\frac{2}{3}t...
  9. Metaphysics

    Find time from distance-speed graph

    In audio application i can automate playrate envelope (playhead speed) 1 = original speed, 2 = 2 x original speed, 3 = 3 x original speed etc.. Now i need to calculate time it takes for playhead to travel each section its easy to calculate time for section 1, where speed is constant time =...
  10. L

    MHB Graph: showing that diameter is greater than average pairwise distance

    I need to prove the above statement. I have a very strong gut feeling that the above equation is not true, and so I need to find a case where the graph diameter is greater than the average pairwise distance. First off, I would like to clarify about the average pairwise distance, which is given...
  11. Athenian

    Finding the Magnetic Field (B_0) on a Graph [NMR]

    Background Information (Not Strictly Necessary): As a quick recap, the graph I am dealing with is a semi-logarithmic graph of free induction decay (FID) amplitude as a function of time. To acquire the value for ##{T_2}^*## (and its uncertainty) in the graph, I used the below equation to do so...
  12. S

    Shape of graph of torque vs angular displacement of galvanometer

    I know the torque will be zero when the deflection is zero and will be maximum when the angular displacement is θ0 but how to determine the exact shape of the graph? Thanks
  13. L

    MHB Prove the minimum and maximum edges in a graph

    Hello! I am having trouble solving the right part of the inequality. For left part of the inequality $n-k \le m$, here’s how I did it Let $ n = v_{1} + v_{2} + v_{3}...+v_{k}$, the sum of vertices of each component in G least number of edges = $(v_{1}-1) + (v_{2}-1) + (v_{3}-1)...+(v_{k}-1)$...
  14. guyvsdcsniper

    What are the units for the slope in an Acceleration vs Mass Graph?

    I am doing an Atwood Simulation lab. I am having trouble with this part of my lab that ask the question provided. I can tell that the relationship is linear and that as the mass increases at a constant rate the acceleration increases. Logger pro tells me that the slope is 31.43. I am having...
  15. M

    MHB Word Problem for slope intercept graph equation

    I have this word problem that is asking for two different answers, the equation for the data and to calculate the shipping rate. I'm not understanding how to address either of the questions. Will someone please help me with this answer?
  16. A

    B 4-colours theorem can be proved visually

    The 4-colour theorem states that the maximum number of colours required to paint a map is 4. The proof requires exhaustive computation with a help of a computer. But I thought that one can visually prove the theorem in the following way; If one replaces the map with a graph where each region...
  17. S

    Graph of torque against displacement of galvanometer needle

    My guess is (c) since the rest of the options do not make sense to me but I am not sure about the shape of the graph. Is (c) the correct answer? If yes, how to know whether the shape will be like in the option (straight line then curve downwards) or other shape like straight line only (like y =...
  18. chwala

    Find the distance traveled on the velocity-time graph

    now this is my approach, my interest in on part b distance travelled= area under the graph between time ##t=0## and ##t=50##, area = ##0.5(40+20)10+ (20 × 40) + 0.5(40+20)10+ 0.5×10×20## Area= ##300+800+300+100=1500m## now textbook says i am wrong, where have i made a mistake?
  19. chwala

    Finding distance on a distance-time graph

    this question caught me by surprise, i had made an assumption that its straight forward...my interest is on question 1.b the total distance traveled is ##1+1+8=10##... average speed is therefore = ##2.5m/s##
  20. B

    Find an Interesting Topic in Graph Theory for an Exam

    Hello everybody, I hope it is the right section to post. For an exam, I should delve into a topic concerning graph theory. My work should include theoretical explanation, pseudo code, correctness analysis, complexity analysis and code implementation (C ++, python or other). Could someone...
  21. dahoom102

    Getting the angular velocity using the angular acceleration graph

    The answer here is A What i did is getting the area as follows, 2×4×1/2 +3×-6×1/2 +4×-6 = -29 and then use this Δω=ωf-ωi -29=ωf-5 ωf=24 but there is no such choice.
  22. Discordy

    Studying Connected Papers: a visual tool to find and explore academic papers

    Hi everyone! We launched Connected Papers 7 months ago, with the goal to help researchers visually find and explore academic papers. Input: a paper of your liking. Output: a full interactive graph of similar papers to explore. For example, here is the graph for The Coevolution of Galaxies and...
  23. Username600

    Help with Creating An Excel Line Graph

    -------------------------------- Hi, just some backstory: My mom and my brother Rob, both loaned me some money in August 2020. They each loaned me $1000.00. I decided to use excel spreadsheet to keep track of my progress paying them both back. Okay so here is the data table I created; as you...
  24. person123

    MATLAB Computing Sections of a Graph Which Can Be Removed By Removing Two Edges

    Hi. I have graph which I am analyzing in MATLAB. I would like to determine sections of the graph which could be removed from the rest of the graph by removing only 2 edges. (The graph represents the network of atomic bonds from a simulation of silica, and I am attempting to locate strands of...
  25. E

    A Can an Elephant Be Graphed? A Look at Von Neumann's Approach

    Here is an elephant. :smile:
  26. seel

    Combining Graphs: Finding Solutions through Graph Homorphisms

    We are working on an architectural project using graphs and we are looking for a way to combine separate graphs into one. We looked into methods to generate graph homorphisms, following the maximum graph homomorphism (MGH) (Langberg et al. 2006) and Paweł Rzążewski’s Exact Algorithm (2014)...
  27. sahilmm15

    B Graph of ax+by=c: How a,b,c Affects Graph

    How does a,b,c effects the equation ax+by=c graphically. Thanks!
  28. L

    Physics Graph Word Problem -- Motion of a person skiing down a slope

    I was able to do the first graph knowing that acceleration is 9.8 and my distance goes up by 10s (y-axis) and my time goes up by 1s (x-axis). For the other 3 graphs, I'm not sure where to begin because I don't know how to figure out my velocity
  29. shk

    Force extension graph for copper wire

    I have attached the homework with some of the solution. I have ciuple of questions about it. For part b, what else I can say? I think the copper wite is ductile . Elastic and plastic behaviour have already been mentioned. I need to add another word to describe it. Can say it's hard ? Or...
  30. F

    Find the Limit of the Heaviside Step Function Added to this Graph

    Here's my attempt: I would like to confirm my answers.
  31. R

    Engineering Spinodal and Isotherms graph on P-V diagram

    Hey guys, During my work I have to show on a P-V diagram certain points on the spinodal for water. How do I draw the spinodel and isothermal lines on a diagram (for example in Excel)? Is it something that I need to calculate? I'm lost on this, I did not find anything relevant on the internet...
  32. B

    B How to Linearize an R vs θ graph (distance vs angle)

  33. brotherbobby

    Which points on the plane satisfy the given equation?

    Given the equation : ##|y| x = x##. Two conditions are possible : (1) ##\underline{y\geq 0}## : ##xy = x\Rightarrow \boxed{y = 1}\; (x \neq 0)##. We note that except for zero, ##-\infty<x<+\infty## for this case. (2) ##\underline{y < 0}## : ##-xy = x\Rightarrow \boxed{y = -1}\; (x \neq 0)##...
  34. W

    MHB Angle at Y-axis Crossing of y=sin(1/x) Graph

    at which angle does the graph y=sin(1/x) cross the y-axis??
  35. SadPaul

    Comp Sci Shortest path problem in bipartite graph

    Hey guys, I need a little help with this exercise so I know I'm on the right path. My explanation: The Bellman-Ford-Moore algorithm computes shortest paths in O(nm) time, so in this situation we say that in a directed bipartite graph the number of iterations that the algorithm will do is...
  36. chwala

    Finding the median in a bar graph problem

    i do not understand the language used in part a, otherwise i would indicate the number of children the women have in order as ##1,2,3,4,5## and ##6## now for part b) this is where i have a problem. To find the median assumably for the number of children, we have to find the total number of...
  37. Marso

    Multiple Questions — Evaluating the graph of this polynomial

  38. shk

    Motion problem -- Calculations given the v(t) graph of a person jumping

    Hi I have answered the question and have attached my working . I'm not very sure about part b. I generally need someone to check my working and let me know if I've made a mistake. Thanks
  39. Baums Mizushala

    Point on a graph nearest to the origin

    The Attempt at a Solution I know the answer is supposed to be ##(-1,0)##. However when I differentiate the above expression I get. $$ 2x+{\frac 5 2} $$ Then the shortest distance would be when the expression equates to 0. $$ 2x+{\frac 5 2}=0 $$ I should be getting ##x=-1## but solving for ##x##...
  40. A

    Doubt in plotting graph between small and large values

    how to plot these points in the same graph and find the foward biasand reverse bias dynamic resistance for the SILICONE DIODE IN4001 FORWARD BIAS REVERSE BIAS
  41. karush

    MHB -apc.4.2.9 graph of f(x) to g(x)

    image due to graphsok just by observation I chose D since integrals tend to introduce curves not eliminate them and the slope was positive
  42. P

    Understanding a Velocity-Time Graph

    Summary:: I think we are still in the earlier parts of Physics and I am confused at how "values" work for a velocity-time graph. We are using the formulas to solve an area of a triangle and rectangle to find the total displacement. If a diagonal line begins from above and continue to go down...
  43. karush

    MHB 2.1.4 AP calc exam graph properties

    screenshot to avoid typos I picked B just could see the others as definite insights?
  44. AN630078

    Photoelectric Effect Graph and Work Function Questions

    1. a)I have plotted the graph on desmos and attached an image here. b i. The threshold frequency is equal to the x-intercept ~ 5.6*10^14 Hz ii. The work function is equal to the y-intercept ~ -3.75*10^19 J (would it be correct to state that this value is negative?) c. Convert to eV; 3.75*10^19...
  45. jaychay

    MHB Determine the relative maximum and minimum on the graph

    Given that f is the function on (−∞, ∞) and the graph is the derivative of f 1.) Find the critical point on the graph ? 2.) Find the interval of the increasing function on the graph ? 3.) Find the interval of the decreasing function on the graph ? 4.) Find the point which is the absolute...
  46. Sabertooth

    Expressing Elliptic Orbitals As Speed Functions.

    Hi everyone:) I have spend a couple of days trying to teach myself the math of orbital mechanics and have been able to generate a model of the orbital path of Haley's Comet, incorporating realistic distances and periods using Kepler's second law & ellipsoid functions. This is a GIF of the motion...
  47. A

    MHB Can someone help me figure out what to mark off on this graph?

    Can someone help me figure out what to mark off on this graph? I am a beginner in Algebra 2 domain and range functions.
  48. aspodkfpo

    Variation of a graph requiring a proportionality to hold

    Do not understand the statement: Ben’s method requires that the voltage output be directly proportional to the intensity, which it is not. https://www.asi.edu.au/wp-content/uploads/2015/03/2010_Physiscs_solutions.pdf My thoughts are that by I= Io cos(theta)^2 We can relate voltage to theta...
  49. Frigus

    B Why the derivative of area is related to the graph of the function

    the explanation about the question I got from internet is, A very small change in area divided by the dx will give the function of graph so anti-derivative of function of graph should be equal to the area of the function. It also seem quite obvious to me but I am not satisfied by it, It seems to...
  50. Spinnor

    Center of Isaias passes near, pressure drop graph local weather

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