What is Graphs: Definition and 654 Discussions

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. D

    What are the equations of these free fall/drag graphs?

    Homework Statement The screenshots of the graphs are attached. The first graph is a velocity graph of a basketball with respect to time, and it is assumed that it has been launched at an angle of 45 degrees and an initial velocity of 10 m/s. Due to air resistance, which I was able to include...
  2. J

    Which graphs represent different types of current flow?

    Homework Statement (1) Name the waveform(s) that represent a periodical current flow. (2) Name the waveform(s) that represent a direct current. (3) Name the waveform(s) that represent a non-periodical alternating current. (4) Name the waveform(s) that represent a current with a non-positive...
  3. F

    Hi, I have a quick question about graph transformations.

    One of my homework questions said "Explain how to obtain f(x)=-(3+x)^2+1 from the graph of y=x^2." I know somehow you need to move the graph right 3, reflect about the x-axis, and move up one, but I don't know how to factor and manipulate the equation to show this.
  4. opus

    B Understanding Polar and Cartesian Graphs: Comparing Two Coordinate Systems

    Please see the attached image. To my understanding, there are two ways to graph a trigonometric function. One is in the Cartesian Coordinate Plane where we have the values (x,y). The other is in the Polar Coordinate system where we have the values (r,θ). In regards to the image that I've...
  5. opus

    Analyzing the graphs of Greatest Integer Functions

    Homework Statement Consider ##u\left(x\right)=2\left[\frac{-x}{4}\right]## (a) Find the length of the individual line segments of the function, (b) Find the positive vertical separation between line segments. Homework Equations The output of Greatest Integer Functions are always integers. The...
  6. Z

    MHB Why I Get X & Y Mixed Up When Plotting Graphs

    Ok so I don't know what is wrong with me, but when it comes to graphs I often get x and y mixed up and the whole thing becomes very confusing. I had this question plot a graph using y = 2/3x The answer is: if x = 3 y = (2/3 * 3) = 2 x = 3, y = 2 and these are integers which can be plotted...
  7. JorgeM

    C/C++ Is it possible to make graphs using C++?

    I am trying to graph some functions but I am not sure about if it is possible using C/C++ I use some numerical metods trying to interpolate data of much complex functions and get the graph (for example the Riemman Dseta function). But I am not even sure If there does exist one specific...
  8. Mr Davis 97

    I Prove that complete graphs are subgraphs

    I can see intuitively that each complete graph ##K_n## is a subgraph of complete graph ##K_m## when ##m \ge n##. What would a rigorous proof consist of? This is just out of curiosity.
  9. Mr Davis 97

    I Explanation for parity difference for # of simple graphs

    http://oeis.org/A000088 This is a list that gives the number of simple graphs on n unlabeled vertices. Could someone conversant in graph theory explain why the number of simple graphs on 4 unlabeled vertices, which is 11, is the only one that seems to be odd (nontrivially), while the rest seem...
  10. Adhil

    MHB Finding the shortest Hamiltonian cycle in a complete graph

    [SOLVED] Okay so I have an assignment to do with only 1 question left incomplete because we have not covered this section yet, non the less it is due soon and I can't find any helpful resources on how to do this question. I know it has something to do with graphs but I'm still lost. So if...
  11. Const@ntine

    Potential Difference/Voltage-Graph Help Needed (Long)

    Homework Statement Alright, this is a Lab Exercise, so it's light on an actual introduction and data, but I'll do my best to put into words the whole thing. The gist of it is that we have a Vat with water, a millimetre paper at the bottom, and we put two Equipotential surfaces inside. We charge...
  12. LonelyElectron

    Connection between V vs I graphs and Ohm's Law

    Hi all! Wanted to know if this is along the right lines, or if I'm missing anything... Any help is greatly appreciated :) 1. Homework Statement Explain the connection between a graph of potential difference versus current, and Ohm’s law. Homework Equations slope = rise/run V=IR R=V/I The...
  13. A

    Quick Question about Pendulum Graphs

    Hi everyone. I just have a very quick question regarding position time graphs and velocity time graphs for a pendulum. As we know, at the maximum displacement, the acceleration is at its maximum and the velocity is zero, and vice versa when the displacement is zero. When we put this...
  14. D

    B Do Parabolas and Other Non-Linear Graphs Have Slope?

    So the slope is of course a ratio of the change in y-coordinates to the change in x-coordinates. This is easy to see with a linear equation. I just came across a cool math simulator ( https://phet.colorado.edu/sims/equation-grapher/equation-grapher_en.html), and I left the first value (ax^2)...
  15. J

    Gradients of Graphs with and without Rates of Change

    Homework Statement Determining which graphs technically use gradients and which do not. This is not a homework question but will help me with it so I put it here. My instructor said you have to be careful whether or not to call what you are calculating from the graph as the "gradient" or not...
  16. D

    Solving Systems of Linear Equations in Two Variables- Graphs

    Homework Statement Solve the system of equations: { (1/2)x-y=3 and x=6+2y Homework Equations NA The Attempt at a Solution The solution is 3=3, which is an identity, which means that there is an infinite amount of solutions to the system. Here's where my question lies (asked my teacher but...
  17. lekh2003

    Graphs of Current through a Transformer

    I have created a make-shift transformer using a circular core and some wire: It is quite basic, but I have some questions about the currents induced when one side of this transformer has a 220 V AC current passed through it. I have an oscilloscope from which I have the graph: The perfect...
  18. EF17xx

    The Resonating Wineglass Problem: Understanding the Equation and Variables

    I have done an experiment where I vary the levels of water in a wineglass and record the frequencies it produces which differ depending on the height of the water. Below you see 1 equation, apparently I can use this equation to mathematically prove my results from my experiment, (this equation...
  19. S

    Slope of Force vs Frequency^2 and Radius vs Period^2 Graphs

    Homework Statement An experiment that involved swinging a mass in a circle was conducted. After graphing both sets of data, I obtained linear graphs of which I calculated the slopes for. I got a slope of 3.5 for the force vs frequency^2 graph and a slope of 0.73 for the radius vs period^2...
  20. J

    Help with velocity-time graphs

    Homework Statement I have attached the homework question as it is on a graph so easier to see as attachment. I am stuck with last last part about sketching onto the graph the actual correct motion of the parachute Homework Equations None The Attempt at a Solution I have attached my attempt...
  21. H

    B Explanation of what a parameter is

    Please could someone explain, pretty simply, what a parameter and also its relation to graphs? And if possible give a couple of examples. I have trawled through the internet and can't find anything which I understand. Thank you very much.
  22. K

    Graphing Derivatives: How to Find Maxima, Minima, and Points of Inflection

    Homework Statement Only 15 Homework Equations First derivative=maxima/minima/vertical tangent/rising/falling Second derivative=points of inflection/concave upward-downward The Attempt at a Solution $$x=y^3+3y^2+3y+2~\rightarrow~1=3(y^2+2y+1)y'$$ $$y'=\frac{1}{3(y+1)^2}>0,~y\neq...
  23. A

    B How can I approach drawing v-x graphs for projectile motion problems?

    Hello sir, I'm a school going student, and this is my first post here. I'm having problems drawing v vs x graphs. I'm unable to understand what should be my general approach while trying to draw v-x graphs. (I've no problem drawing x-t, v-t, or a-t graphs). Let's say, for example, a ball is...
  24. S

    Determining Relationship Between Two Variables

    Homework Statement An experiment was conducted to determine the relationship between metre-stick projection from a table (L) on the vertical depression of the free end (y). Using the collected data, graphs had to be plotted until a linear relation was found, and the equation of the line had to...
  25. S

    Finding Proportional Relationships using Log-Log Graphs

    Homework Statement [/B] Given a a log-log graph with 8 lines, you must determine the equation of each line in its original relationship. The slope of the graph (m) gives the power of the original relationship. Examples: if m = 2, 3; then y ∝ x^2, x^3, etc. if m = -1, -2; then y ∝ 1/x...
  26. M

    MHB Questions regarding writing equations when given graphs

    Hello, I'm studying for a test this Friday and I have two problems that have completely stumped me. I'm not sure how to solve either. Any help would be greatly appreciated.
  27. W

    Log-Plots: Solving Homework Equations

    Homework Statement Homework EquationsThe Attempt at a Solution I have been trying to rearrange the equation to look something like ln(y) = Aln(x) but to no avail. Something tells me that that's not necessarily the right way to approach the problem, could anyone point out what I should be...
  28. M

    MHB Graph Grammer Function For Two Exotic Graphs Question please.

    Hallo I will start my question by describing a particular graph , 1. There are a finite , fixed, number of nodes. labeled {A,B...Z} (so for this example there are 24 nodes.) 2. Each node consists of another graph (tree) with exactly 3 nodes in 3 depths. That means we have a graph where...
  29. J

    How Do You Graph Kinetic Energy Against Time and Distance?

    <Moderator's note: Moved from a technical forum and thus no template.> A horizontal force of 80 N acts on a mass of 6 kg resting on a horizontal surface. The mass is initially at rest and covers a distance of 5 m in 0.92 s under the action of the force. Assuming there are no energy losses due...
  30. C

    Interpreting I-V graphs in Photoelectric effect

    Homework Statement In photoelectric effect I-V graphs, for the same intensity but with different light frequencies (f2>f1), the I-V graph has the same max photocurrent but |Vs2| > |Vs1|. If we concentrate between the cutoff and the saturation regions of the I-V graph, we can see that at any...
  31. T

    I I would simply like to know how to get 2^k.

    The complete graph K_n can be expressed as the union of k bipartite graphs iff n≤2^k I would simply like to know how to get 2^k.
  32. J6204

    Graphs of y(x,t)=A Sin(kx - wt + φ) are shown below

    Homework Statement The first graph shows y vs t for a point at x=0 m. The second shows y vs x for the string at a time of 2 s. I have calculated the amplitude of the oscillation to be 1.0m I have calculated the angular frequency of the oscillation to be 2.90 rad/s I have calculated the...
  33. Vital

    Graph r = 6 cos() issues with plotting on xy-plane

    Homework Statement Hello! Last week I have came here for the help related to this problem. I am creating a new thread to describe the issue more precisely. I will be grateful for your help and explanation. I post the explanation for the book first accompanied by attached pictures, and below I...
  34. T

    Understanding Digital-to-Analogue Conversion in Signal Processing

    Homework Statement The digitial signal is transmitted and then converted to an analogue form by the DAC. Using data from Fig. 12.2, draw on the axes of Fig.12.3, the output level of the transmitted analogue signal for time zero to time 1.2 ms. Homework Equations Four bit digital numbers...
  35. Vital

    How is the xy curve formed from the rθ curve?

    Homework Statement Hello! I will be grateful for your help in deciphering the meaning of a paragraph from the book. I honestly don't understand how they got the semi-circle on the xy graph by transferring it from rθ graph. Homework Equations I attach the screen shot from the book. The Attempt...
  36. K

    Area between two graphs as a sum

    Homework Statement Homework Equations $$1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}$$ The Attempt at a Solution $$S_n=2\{[f_1(x_1)-f_2(x_1)]\Delta x+[f_1(x_2)-f_2(x_2)]\Delta x+...+[f_1(x_{n-1})-f_2(x_{n-1})]\Delta x\}$$ $$S_n=2\{[18-\Delta x^2-\Delta x^2]+[18-(2\Delta x)^2-(2\Delta...
  37. Vital

    Graphs of sin and cos, how to set points for x values

    Homework Statement Hello! I am at the topic on graphing trigonometric functions. Exercises are rather easy at this point, but I have a problem deciphering how authors of the book choose points for x values. Please, take a look at few examples (including screen shots I attach), and, please...
  38. Bunny-chan

    Understanding Motion Graphs: Converting Units and Interpreting Results

    Homework Statement I'm having some issues with the following exercise: The graph below represents the marking of a vehicle speedometer in funtion of time. Elaborate the corresponding graphs of acceleration and space traveled in function of time. What is the average acceleration of the vehicle...
  39. M

    B Graphs of solids of revolution

    Is drawing of graphs of solids of revolutions important topic of mathematics? This makes me remerber conic sections topic. Conic sections topic belongs to algebra and drawing their graphs is important. So where does solids of revolutions belong to? I know calculation of their volumes belongs to...
  40. Brage Eidsvik

    I Finding Area of Graphs with x^2: A General Solution?

    Hello, If I have an x^2 graph that goes from 0 to a point a. Is there a general solution to where the area of the left side is equal to the area of the right?
  41. M

    MHB Chordal graphs and perfect elimination orderings

    1. G is chordal iff G has a Perfect Elimination Scheme/Ordering (PES) 2. G is chordal iff every induced subgraph H of G has a simplicial vertex in H. I want to show directly (ie, don't use the above) to show that G has a PES iff every induced subgraph H of G has a simplicial vertex. (->)...
  42. T

    Flux and Induced EMF vs displacement graphs

    Homework Statement A small coil of wire is situated in a non-uniform magnetic field. The coil consists of 40 turns of wire and moves with a constant speed in a straight line. The coil has displacement x from a fixed point P. The variation with x of the magnetic flux in the coil is shown in the...
  43. S

    I Explain Graphs -- adiabatic Expansion of mono- di- and poly- gas

    "How it can be explained mathematically by dP/dV = -gammaP/V
  44. S

    Potential Energy vs. Position Graphs

    Homework Statement A particle has the potential energy shown in the figure. (Figure 1) What is the x-component of the force on the particle at x =5, 15, 25, and 35 cm? I'm stuck on finding the Fx at 25 & 35 cm.Homework Equations F = -du/dx 100 N = 1J/cm The Attempt at a Solution Slope...
  45. Poetria

    Complex functions with a real variable (graphs)

    Homework Statement How do the values of the following functions move in the complex plane when t (a positive real number) goes to positive infinity? y=t^2 y=1+i*t^2[/B] y=(2+3*i)/t The Attempt at a Solution I thought: y=t^2 - along a part of a line that does not pass through the...
  46. snoopies622

    Blackbody and gas molecule speed graphs look similar. Why?

    The graphs of intensity versus wavelength of blackbody radiation and of number of molecules versus speed in a gas look very similar to me. Is this just a coincidence? They seem like quite different phenomena.
  47. M

    Interpreting graphs from isentropic efficiency of a compressor lab

    Homework Statement We did a lab experiment for isentropic efficiency of a compressor and was asked to plot the following graphs, change in stagnation pressure across the compressor (∆p0) vs air mass flow rate, and isentropic efficiency vs air mass flow rate. I have plotted the graphs but not...
  48. R

    How can I understand wave graph conversion better?

    Homework Statement Number 6: For this problem I'm changing the wave speed to 3.0 m/s instead of 1 m/s because that's what our teacher instructed us to do. Homework Equations None that I know of The Attempt at a Solution I'm having the hardest time making connections between the graphs and...
  49. A

    Drawing flux and induced EMF graphs - Faraday's/Lenz's law

    I would really appreciate some clarification about how to draw graphs showing changing flux and induced EMF. I understand that there needs to be a 90 degree phase difference as the EMF is a maximum when the flux has the greatest rate of change. However, our exam board says that the EMF can be...
  50. ThatOneMidget

    Having real problems with working out scales for graphs

    I have no clue what I'm supposed to do with the scales for my graphs. I am an AS level student and in the practical paper we're asked to draw a graph using our readings. The marking scheme has this to say: I have no idea what they mean by awkward scales and what not, what i usually do to decide...
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