What is Graph: Definition and 1000 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. bluesteels

    PV Graph Question: Internal Energy Not Zero?

    Figure 1: im so confused on why is the internal energy not zero for the 2nd picture because i thought if the gas returning to its original states so it zero
  2. karush

    MHB Graph of $y=\sin{x}-2$ on the domain $[0,2\pi]$

    Graph $y=\sin{x}-2$ on the domain $[0,2\pi]$ This is a sample math problem in preparation for the entrance exam for the USAF Academy Even not asked I thot also the Period, Amplitude, PS and list some observations that should be know to graph without an app 1. we know that sin(0)=0 so sin(x)...
  3. rudransh verma

    How Does Acceleration Affect Velocity in 1D Motion?

    ##v=\frac12 *1*4+\frac12*1*4= 4 m/s## but the answer is wrong.
  4. rudransh verma

    Understanding Distance and Displacement on Velocity-Time Graphs

    3) and 4) is easy. Average and instantaneous acceleration is same from 1 to 4 sec since it’s constant acceleration. 1) and 2) I am unable to get the correct area under the curve. ##s = \frac 12*(-5)*1 + \frac12*10*2 + 10*1 + \frac12*10*1## Same I guess will be distance. 5) what is the graph...
  5. Andrea Vironda

    Is there something happening below 0°C in this daily temperature graph?

    Hi, I notice from the graph that every night when the temperature reaches -2/-3 °C the curve change first the slope and then the concavity. Is something special happening?
  6. DaveC426913

    B Periodic spiral graph interpretation

    Could someone explain the geometry of this graph? Why does the radial distance vary non-uniformly? To-wit: Distance from origin to Nov 2020 is much larger than Nov 2020 to Nov 2021 Why are there two areas - one above and one below - the centre line...
  7. 0

    Moving the graph to the right -- What do you think?

    I'm studying calculus alone with textbooks. The part about moving the graphs to the right or to the left struck me because they just have a list of rules, properties and make you relate the graph with the corresponding equation. I know what is the rate of change and I thought I could do better...
  8. AntSC

    Acceleration-Time Graph and Velocity Multiple Choice Question

    The variation with time t of the acceleration a of an object is shown What is the change in velocity of the object from ##t=0## to ##t=6##? A. ##6ms^{-1}## B. ##8ms^{-1}## C. ##10ms^{-1}## D. ##14ms^{-1}## So apparently the answer is B, which I am having trouble reconciling. Using methods...
  9. karush

    MHB -7.8.98 amplitude period PS VS graph. of cos eq

    Find amplitude, period, PS, VS. then graph. ok I think these are the plug ins we use $Y_{cos}=A\cos\left[\omega\left(x-\dfrac{x \phi}{\omega} \right)\right]+B \implies A\cos\left(\omega x-\phi\right)+B \implies T=\dfrac{2\pi}{\omega} \implies PS=\dfrac{\phi}{\omega}$ ok I wanted to do...
  10. chwala

    Solve the vertical stretch/compression graph problem

    This is the problem, Let ##y=f(x)= (x-2)^2##. The graph of ##y=af(x)##can be obtained from the graph of ##y=f(x)## by a stretch parallel to the y- axis with scale factor ##a##. In our case here, ##a=3##, therefore the corresponding graph is as indicated in blue. Find my graph below using desmos.
  11. ASKINGHUMAN

    Confused about calculating the average energy from a distribution graph

    Hello, in one of tasks of my liquid scintillation lab is to determine the average energy. You can see from the graph that data I obtained is similar to this one that I have a excel sheet data. X-axis is for beta particle energy from 0-156keV while y-axis counts of the particles. So according to...
  12. karush

    MHB 8.2.2 Int area of a region and graph

    a. Sketch the region of integration and evaluate the Integral b. Evaluate $V=5\displaystyle\int_0^8 \biggr[ x^3\biggr]_{(y-4)/2}^{y^{1/3}}\ dy \ =5\displaystyle\int_0^8 [(y^{1/3})^3-((y-4)/2))^3] \quad \ dy \ =5\displaystyle\int_0^8 \biggr [y-\dfrac{(y-4)^3}{8}\biggr] \ dy$ Expand...
  13. ektov_konstantin

    I Moving center of coordinates in the polar graph

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  14. srfriggen

    How to graph by hand: y=log((x/(x+2))

    I first attempted to find the x and y intercepts, algebraically, and discovered there were none. I then split the equation into y= log(x) - log(x+2) to see if that would give me any insight. It did not. I used a graphing calculator and saw many similarities between x/(x+2) and log((x/(x+2)) but...
  15. B

    I Bias and dark current from a graph of mean signal vs exposure time

    I have a graph of mean signal (per pixel) vs exposure time (sec) for 8 different dark frames. I am being asked to find the bias in ADUs/pixel and the dark current in ADUs/sec/pixel and I am very confused on how I could get it. I know that the average of all of the mean signals is a rough guide...
  16. mcas

    Estimate the following quantities from a graph of the refractive index of NaCl

    a) I managed to obtain some results that are roughly around what is given in the answers. Because \varepsilon_{st} and \varepsilon_{\infty} are values of \varepsilon_{1}, I used this approximation: n\approx \frac{1}{\sqrt{2}} (\varepsilon_{1}+\sqrt{\varepsilon_{1}^2})^{1/2} -> \varepsilon_{1} =...
  17. Bhawna L

    Sketching a graph of contaminants after a train crash

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  18. vpt069

    I just wanted to know how to find error/uncertainties from the graph

  19. K

    I Curvature of 3D Graph on Point w/ Directional Vector

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  20. I

    How can I plot the function g(x) = sin(πn/L) x and its corresponding g²(x)?

    Summary:: We are currently studying basics of quantum mechanics. I'm getting the theory part but it's hard to visualise everything and understand. We are given this question to plot the function so if someone could help me in this. Plot the following function and the corresponding g²(x) g(x)...
  21. K

    A Geodesics on three dimensional graph

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  22. B

    I Equation to graph a sine wave that acts like a point on a unit circle

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  23. C

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  24. R

    Calculate quality factor of a damped oscillation from a graph

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  25. G

    Need help with the interpretation of this v(t) graph

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  26. karush

    MHB B.2.2.16 ok wanted of modify this graph to be this......

    \begin{tikzpicture} [declare function = {radius(\phi)=sqrt((3*sin(\phi)+cos(\phi)) / (sin(\phi)^3 -cos(\phi)^3)); },] \draw[->] (-3,0) -- (3.4,0); \draw[->] (0,-2) -- (0,4.2); \draw[domain=50:{atan(-1/3)+179.99}, variable=\phi, red, smooth, ultra thick] plot ({radius(\phi) * cos(\phi)}, {1 +...
  27. karush

    MHB Normal distribution graph P(a<x<b)

    The probability that the lifespan of an insect of this species lies between 55 and 60 hours is represented by the shaded area in the following diagram.\\ Write down the values of a and b. $a=\dfrac{2}{4.4}= 0.455 b=\dfrac{3}{4.4}=0.682]$ ok this was a key to a test question from 2013 but mostly...
  28. J

    I Is this business graph an exponential or polynomial function?

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  29. S

    B Question about stress - strain graph and definition of hardness

    This is from my notes: Point D is called ultimate tensile strength and defined as highest possible within this material. So it means that point D should be at the highest point of the graph (more like absolute maximum in math)? Because it seems that from the graph point D is not at maximum...
  30. S

    Graph of waveform across diode shown by CRO

    Option (A) and (B) is wrong because the waveform should be half-wave, not full-wave. But how to know whether it will be (C) or (D) based on the circuit given? Thanks
  31. karush

    MHB What is the graph of the function with an initial value problem of 2.2.21?

    this had ahttps://mathhelpboards.com/threads/2-2-21-ivp.27772/ but wanted to add tikz graph orifinally authored by Klazs van Aarsen \begin{tikzpicture}%[scale=.6] [declare function = {radius(\phi)=sqrt((3*sin(\phi)+cos(\phi)) / (sin(\phi)^3 -cos(\phi)^3)); },] % \draw[help lines] (-3,-2) grid...
  32. M

    I Graph Representation Learning: Question about eigenvector of Laplacian

    Hi, I was reading the following book about applying deep learning to graph networks: link. In chapter 2 (page 22), they introduce the graph Laplacian matrix ##L##: L = D - A where ##D## is the degree matrix (it is diagonal) and ##A## is the adjacency matrix. Question: What does an...
  33. jaychay

    MHB Complex number equation graph problem

    Given (a,b) is the coordinate just like (x,y). Find equation Zo and coordinate (a,b) ?Please help me Thank you in advance.
  34. B

    Evaluate the complexity of this graph coloring algorithm

    Hello everybody, I should evaluate the complexity of this graph coloring algorithm. To do this, I use the adjacency matrix in which the graph nodes are the elements on the diagonal, while the elements outside the diagonal indicate if a node is adjacent to another ##(A_{i,j} = 1)## or not...
  35. karush

    MHB 7.8.11 Find amplitude, period, PS, VS. graph 2 periods

    $\tiny{\textbf{7.8.11 Campbell HS}}$ Find (A)mplitude, (P)eriod, PS, VS. graph 2 periods $y=3\cos(\pi x-2)+5$ by observation we have A=3 and VS=5 ok assume $\omega=\pi$ so if period is $T=\dfrac{2\pi}{\omega}$ then $T=\dfrac{2\pi}{\pi}=2$
  36. karush

    MHB 7.8.99 find PS, VS Period, graph

    $\tiny\textbf{7.8.a09 Radford HS}$ Find amplitude, period, PS, VS. then graph. $y=\cos\left(x+\dfrac{\pi}{2}\right)$For the graphs of $y=A\sin(\omega x - \phi)$ or $y=A\cos(\omega x - \phi),\omega>0$ Amplitude $=|A|$ Period $T=\dfrac{2\pi}{\omega}=\dfrac{2\pi}{2}=\pi$ PS...
  37. B

    Sudoku solver by graph coloring theory

    Hello everybody! I have to implement a sudoku solver in C ++ taking advantage by graph coloring theory, where each number to insert is a color of the associated graph node. In particular I would like to use the Welsh-Powell algorithm. I find myself in trouble starting with this project and I...
  38. G

    Velocity-distance graph for a freely falling body

    My reasoning: Gradient of graph = v/s = t t increasing therefore gradient increasing So graph B The answer is C.
  39. N

    Use Graph To Investigate Limit

    Use a graph to investigate limit of f(x) as x→c at the number c. Note: c is given to be 2. This number comes from the side conditions of the piecewise function. See attachments. lim (x + 2) as x tends to c from the left is 2. lim x^2 as x tends to c from the right is 4. LHL does not...
  40. N

    Use the graph of f(x) to investigate the limit

    Use the graph to investigate the limit of f(x) as x tends to c at the number c. See attachments. Based on the graph of f(x), here is what I did: lim (2x + 1) as x tends to 0 from the left is 1. lim (2x) as x tends to 0 from the right is 0. LHL does not equal RHL. I conclude the limit of...
  41. N

    Investigating A Limit Via Graph

    Use the graph to investigate (a) lim of f(x) as x→2 from the left side. (b) lim of f(x) as x→2 from the right side. (c) lim of f(x) as x→2. Question 18 For part (a), as I travel along on the x-axis coming from the left, the graph reaches a height of 4. The limit is 4. It does not matter if...
  42. N

    Use Graph To Investigate Limit

    For questions 24 and 26, Use the graph to investigate limit of f(x) as x→c. If the limit does not exist, explain why. Question 24 For (a), the limit is 1. For (b), the limit is cannot be determined due to the hole at (c, 2). For (c), LHL does not = RHL. I conclude the limit does not exist...
  43. N

    Use Graph to Determine Limit: Calculating Limits with Piecewise Functions

    Summary:: Graphs and Limits Use the graph to determine the limit of the piecewise function as x tends to 1. Let me see. lim of (-x + 3) as x-->1 from the left is 2. lim of (2x) as x-->1 from the right is 2. I can safely say that the limit of f(x) as x tends to 1 from the left and right...
  44. 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...
  45. 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 $y=1$
  46. 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...
  47. 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...
  48. 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.
  49. 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...
  50. 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...
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