Modeling with tables, equations, and graphs (article) | Khan Academy
The relationship between a and F in the previous example is linear. When you see the word linear in physics, think 'line'. A graph of two. We use the cartesian coordinate system to graph functions. (this is how we express relationships between quantities) and their graphs. In order to graph a linear equation we work in 3 steps: First we solve the equation for y. Second we make a table for our x- and y-values. From the x values we.
Linguistics[ edit ] Graph-theoretic methods, in various forms, have proven particularly useful in linguisticssince natural language often lends itself well to discrete structure. Traditionally, syntax and compositional semantics follow tree-based structures, whose expressive power lies in the principle of compositionalitymodeled in a hierarchical graph.
More contemporary approaches such as head-driven phrase structure grammar model the syntax of natural language using typed feature structureswhich are directed acyclic graphs.
Within lexical semanticsespecially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; semantic networks are therefore important in computational linguistics. Still, other methods in phonology e. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as TextGraphsas well as various 'Net' projects, such as WordNetVerbNetand others.
Physics and chemistry[ edit ] Graph theory is also used to study molecules in chemistry and physics.
Graph theory - Wikipedia
In condensed matter physicsthe three-dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. Also, "the Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand.
This approach is especially used in computer processing of molecular structures, ranging from chemical editors to database searching. In statistical physicsgraphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.
Similarly, in computational neuroscience graphs can be used to represent functional connections between brain areas that interact to give rise to various cognitive processes, where the vertices represent different areas of the brain and the edges represent the connections between those areas. Graph theory plays an important role in electrical modeling of electrical networks, here, weights are associated with resistance of the wire segments to obtain electrical properties of network structures.
Chemical graph theory uses the molecular graph as a means to model molecules. Graph theory in sociology: Moreno Sociogram  Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore rumor spreadingnotably through the use of social network analysis software.
Under the umbrella of social networks are many different types of graphs. Influence graphs model whether certain people can influence the behavior of others. Finally, collaboration graphs model whether two people work together in a particular way, such as acting in a movie together. Biology[ edit ] Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist or inhabit and the edges represent migration paths or movement between the regions.
This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species.
Mathematics[ edit ] In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Algebraic graph theory has close links with group theory. Other topics[ edit ] A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphsare used to represent structures in which pairwise connections have some numerical values.
Graph functions and relations
For example, if a graph represents a road network, the weights could represent the length of each road. There may be several weights associated with each edge, including distance as in the previous exampletravel time, or monetary cost. Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy  and L'Huilier and represents the beginning of the branch of mathematics known as topology.
The techniques he used mainly concern the enumeration of graphs with particular properties. These were generalized by De Bruijn in Cayley linked his results on trees with contemporary studies of chemical composition.
In particular, the term "graph" was introduced by Sylvester in a paper published in in Naturewhere he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams: Many incorrect proofs have been proposed, including those by Cayley, Kempeand others. The study and the generalization of this problem by TaitHeawoodRamsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus.
The four color problem remained unsolved for more than a century. In Heinrich Heesch published a method for solving the problem using computers.
A simpler proof considering only configurations was given twenty years later by RobertsonSeymourSanders and Thomas. Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra.
The first example of such a use comes from the work of the physicist Gustav Kirchhoffwho published in his Kirchhoff's circuit laws for calculating the voltage and current in electric circuits.
Graph drawing Graphs are represented visually by drawing a dot or circle for every vertex, and drawing an arc between two vertices if they are connected by an edge.
If the graph is directed, the direction is indicated by drawing an arrow. A graph drawing should not be confused with the graph itself the abstract, non-visual structure as there are several ways to structure the graph drawing.
All that matters is which vertices are connected to which others by how many edges and not the exact layout. In practice, it is often difficult to decide if two drawings represent the same graph. Depending on the problem domain some layouts may be better suited and easier to understand than others.
The pioneering work of W. Tutte was very influential on the subject of graph drawing. Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings.
Graph drawing also can be said to encompass problems that deal with the crossing number and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a planar graphthe crossing number is zero by definition. Drawings on surfaces other than the plane are also studied. Graph-theoretic data structures[ edit ] Main article: Graph abstract data type There are different ways to store graphs in a computer system.
The data structure used depends on both the graph structure and the algorithm used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. Inverse Relationship Now, let's look at the following equation: Note that as X increases Y decreases in a non-linear fashion.
This is an inverse relationship. Example of an inverse relationship in science: When a higher viscosity leads to a decreased flow rate, the relationship between viscosity and flow rate is inverse. Inverse relationships follow a hyperbolic pattern. Below is a graph that shows the hyperbolic shape of an inverse relationship. Quadratic formulas are often used to calculate the height of falling rocks, shooting projectiles or kicked balls.
A quadratic formula is sometimes called a second degree formula. Quadratic relationships are found in all accelerating objects e. Below is a graph that demostrates the shape of a quadratic equation. Inverse Square Law The principle in physics that the effect of certain forces, such as light, sound, and gravity, on an object varies by the inverse square of the distance between the object and the source of the force.
In physics, an inverse-square law is any physical law stating that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity.
The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.
One of the famous inverse square laws relates to the attraction of two masses. Two masses at a given distance place equal and opposite forces of attraction on one another. The magnitude of this force of attraction is given by: The graph of this equation is shown below. More on Brightness and the inverse square law Damping Motion Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations.
In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators.
Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems. Sine Wave Relationship The graphs of the sine and cosine functions are sinusoids of different phases.
The sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. It is named after the function sine, of which it is the graph.
Lab Activities and Resources What are Mathematical Relationships What is a mathematical relationship and what are the different types of mathematical relationships that apply to the laboratory exercises in the following activities.