Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. In one restricted but very common sense of the term, a graph is an ordered pair comprising: a set of vertices (also called nodes or points); a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with two distinct vertices). To avoid ambiguity, this type of object may be called precisely an undirected simple graph. In the edge , the vertices and are called the endpoints of the edge. The edge is said to join and and to be incident on and on . A vertex may exist in a graph and not belong to an edge. Multiple edges, not allowed under the definition above, are two or more edges that join the same two vertices. In one more general sense of the term allowing multiple edges, a graph is an ordered triple comprising: a set of vertices (also called nodes or points); a set of edges (also called links or lines); an incidence function mapping every edge to an unordered pair of vertices (that is, an edge is associated with two distinct vertices). To avoid ambiguity, this type of object may be called precisely an undirected multigraph. A loop is an edge that joins a vertex to itself. Graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph) which is not in . So to allow loops the definitions must be expanded.

Ontology

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Related exercises (22)

Graphs

The position as a function of time of a car in linear motion is given by

\begin{cases} \displaystyle{\frac{1}{2}a_0t^2} \qquad\quad\ \,\, \text{if} \quad t\leq \displaystyle{\frac{v_0}{a_0}} \\ v_0t - \displaystyle{\frac{v_0^2}{2a_0}} \qquad \text{if} \quad t> \displaystyle{\frac{v_0}{a_0}} \end{cases}$$ where v₀ and a₀ are positive constants. The position of the car is illustrated on the graph below: 1. Draw the graphs of velocity v(t) and acceleration a(t) as a function of time. 1. The po

Current induced by another current loop

The current through the coil on the right in Fig.xa0fig:fig7 varies as shown in Fig.xa0fig:fig7a. Which graph in Fig.xa0fig:fig7b best shows the ammeter reading as a function of time? The field created by the first loop:

first, it constantly grows with time ⟹ should induce a constant current in the second loop;
then it’s constant ⟹ no current in the second loop;
finally, it constantly drops with time ⟹ should induce a constant current in the opposite directi

Round trip

A vehicle is in linear motion. The scalar velocity as a function of time is given by v(t)u2004=u2004b₀t²−u2006a₀txa0, where b₀ and a₀ are positive constants.

Determine the position x(t) of the vehicle as a function of time knowing that it is initially at the origin, i.e. x(0)u2004=u20040.
For b₀u2004=u20041 and a₀u2004=u20041, draw the graphs of the position x(t) and of the velocity v(t).
For b₀u2004=u20041 and a₀u2004=u20041, what happens at times tu2004=u20041 and $t=\frac{3}{2}

Perforated plate

We consider a square plate with side of length a and uniform density. We pierce a circular hole of radius R in the plate such that the distances between the hole’s edge and two adjacent edges of the plate are b and c, with b,u2006cu2004

Spherical shell

Calculate, in any point of space, the electrostatic potential

V(\bf{r})

and the field

\bf{E}(\bf{r})

created by a spherical shell uniformly charged with an internal radius a, external b, and of total charge u2005+u2005Q, centered at the origin O of the cartesian axes. Graphically represent the functions

V(\bf{r})

and

E_r(\bf{r})

(the radial component of

\bf{E}(\bf{r})

) Replace now the charged shell by an ideal conductor of similar shape. The conductor is neutral and isolated. A charge u2005+u2005

Related notebooks (25)

Powder X-Ray Diffraction Analysis: Exploring Crystal Structures.

XRay Diffraction[1]¶ The purpose of this notebook is to study the powder XRay diffraction spectra of some systems. When the crystal is hit with an XRay wave, the wave is diffracted only for specific case where the Bragg condition is satisfied for one of the lattice planes in the material. For single crystal diffraction, the condition is satisfied only for specific orientation of the crystal, such that the incident ray has the correct

\theta

value for the plane. In powder XRay data, however, any plane ori

Density of States in 1D Systems

This notebook shows density of states in 1D. Note that all graphs are cut off at Fermi Energy Ef, hence if the cut is visible on the graph consider increasing number of electrons in the system."

Interactive Phonon Band Structure Visualization

In [1]: import numpy as np import matplotlib.pyplot as plt from matplotlib import animation from matplotlib import rc import ipywidgets as ipw from ipywidgets import interact,interactive import ipywidgets as widgets from IPython.display import HTML %matplotlib widget global global_K1 global global_K2 global global_branche global global_selected_k In [2]: def get_closest_point(clic_x,clic_y): # Get the data from the figure ax= plt.gca() # get axis handle acoustic_data = ax.lines[1] op

Estimation of Electron Density and Mean Free Time for Transparent Conductors

Chapter 1: Transparent Conductors[1]¶ Instruction: The purpose of this notebook is to try and find values of electron density and mean free time for a material suitable to be used as a transparent conductors. The final estimate should be the order of the two parameters to be used[2]¶ In [1]: import numpy as np from bokeh.io import push_notebook, output_notebook, show from bokeh.layouts import row, column from bokeh.plotting import figure output_notebook() from ipywidgets import interact from collect

Temperature Dependence of the Fermi-Dirac Distribution

This notebook serves to show how the Fermi-Dirac distribution and its derivative change with temperature.\n", " \n", "The Fermi-dirac distribution is

f(E)=\\frac{1}{e^{(E-\\mu)/k_bT}+1}

\n", "\n", "For the purpose of simplicity, we will plot the graph of

f(E-\\mu)

. This allows us to avoid complications connected with

\\mu

being a function of temperature (consult the other notebook for this week or section 2.2.3 in the notes for more details)\n", "\n", "We first define the r

Related exercises (22)

Graphs

The position as a function of time of a car in linear motion is given by

\begin{cases} \displaystyle{\frac{1}{2}a_0t^2} \qquad\quad\ \,\, \text{if} \quad t\leq \displaystyle{\frac{v_0}{a_0}} \\ v_0t - \displaystyle{\frac{v_0^2}{2a_0}} \qquad \text{if} \quad t> \displaystyle{\frac{v_0}{a_0}} \end{cases}$$ where v₀ and a₀ are positive constants. The position of the car is illustrated on the graph below: 1. Draw the graphs of velocity v(t) and acceleration a(t) as a function of time. 1. The po

Current induced by another current loop

first, it constantly grows with time ⟹ should induce a constant current in the second loop;
then it’s constant ⟹ no current in the second loop;
finally, it constantly drops with time ⟹ should induce a constant current in the opposite directi

Round trip

A vehicle is in linear motion. The scalar velocity as a function of time is given by v(t)u2004=u2004b₀t²−u2006a₀txa0, where b₀ and a₀ are positive constants.

Determine the position x(t) of the vehicle as a function of time knowing that it is initially at the origin, i.e. x(0)u2004=u20040.
For b₀u2004=u20041 and a₀u2004=u20041, draw the graphs of the position x(t) and of the velocity v(t).
For b₀u2004=u20041 and a₀u2004=u20041, what happens at times tu2004=u20041 and $t=\frac{3}{2}

Perforated plate

Spherical shell

Calculate, in any point of space, the electrostatic potential

V(\bf{r})

and the field

\bf{E}(\bf{r})

V(\bf{r})

and

E_r(\bf{r})

(the radial component of

\bf{E}(\bf{r})

) Replace now the charged shell by an ideal conductor of similar shape. The conductor is neutral and isolated. A charge u2005+u2005

Related notebooks (25)

Powder X-Ray Diffraction Analysis: Exploring Crystal Structures.

\theta

value for the plane. In powder XRay data, however, any plane ori

Density of States in 1D Systems

This notebook shows density of states in 1D. Note that all graphs are cut off at Fermi Energy Ef, hence if the cut is visible on the graph consider increasing number of electrons in the system."

Interactive Phonon Band Structure Visualization

Estimation of Electron Density and Mean Free Time for Transparent Conductors

Temperature Dependence of the Fermi-Dirac Distribution

This notebook serves to show how the Fermi-Dirac distribution and its derivative change with temperature.\n", " \n", "The Fermi-dirac distribution is

f(E)=\\frac{1}{e^{(E-\\mu)/k_bT}+1}

\n", "\n", "For the purpose of simplicity, we will plot the graph of

f(E-\\mu)

. This allows us to avoid complications connected with

\\mu

being a function of temperature (consult the other notebook for this week or section 2.2.3 in the notes for more details)\n", "\n", "We first define the r

Graph theory

Chat with Graph Search

This notebook shows density of states in 1D. Note that all graphs are cut off at Fermi Energy Ef, hence if the cut is visible on the graph consider increasing number of electrons in the system."

This notebook shows density of states in 1D. Note that all graphs are cut off at Fermi Energy Ef, hence if the cut is visible on the graph consider increasing number of electrons in the system."

Ordered Ramsey numbers of graphs with 𝑚 edges

Spectral Clustering with Side Information

Clique density vs blowups