A Graph Is An Abstract Structural Data That Is Implemented As A Directional And Non-Directional Graph And Its Purpose Is To Apply The Concept Of Graph To Mathematics And Especially Graph Theory. 

Famous Graphs, A graph structure data basically consists of a finite set of ordered pairs called edges and units called vertices or nodes.

Just as in mathematics, we say that u goes to v for one edge (u, v) or that u and v are adjacent. In the world of computation, one graph is assigned a number to each edge, in which case a weighted graph exists. Comes.

Free-triangle graph

A free-triangle graph is a directionless graph whose three vertices do not form a triangle. Equivalent to free-triangle graphs can be graphs with a maximum number of clusters of 2, graphs with the smallest circumference of at least 4, graphs without triple rounds, or graphs with local independence.

According to Turan’s theory, an n-free-triangle vertex graph with the largest number of edges in a complete two-part graph in which the number of vertices in each segment is as equal as possible. The problem of finding a triangle is like whether a graph is a free triangle or not.

When a graph has a triangle, algorithms usually need to find three vertices that form a triangle together.

Empty graph

A blank graph may refer to a zero-order graph or is equivalent to a bar graph. (The latter is sometimes referred to as the “empty graph”). In category theory, an empty graph is, by some definitions, a “graph group” of the primary member of the category. The bottom graph K {0 is a unique graph that has no vertices (so the order is zero). As a result, this graph has no edges.

Dense graph

A dense graph is a graph whose number of edges is close to the maximum number of edges. In contrast to dense graphs, a graph is a thin graph with few edges.

Line graph

Consider a non-empty graph G. If we consider a vertex instead of each edge, and we connect the two vertices. Suppose the corresponding edges of the two vertices in G are common at one vertex of G.

Complete graph

A graph is a complement to a graph that is uniform with its complement. Two of its simplest graphs complement a path graph with 4 vertices and a distance graph with 5 vertices. The problem of recognizing the alignment of two graphs and recognizing the self-completeness of a graph is the same as the general problem of graph alignment.

Double bond

A communication network is defective against defects if it has other paths between the heads. The more separate routes, the better. This example is exactly the concept of a k-vertex-connected graph. A biconnected graph is a special case of a multi-link graph. A two-link graph is a link graph with the property that is inseparable. That is, it remains intact by removing one vertex.

This feature is useful for managing a graph with dual redundancy to prevent the graph from unlinking by removing one edge. Due to the redundancy property, the use of dual-band graphs in networks is significant.

Split graph

A split graph is a graph whose vertices can be separated by a single click and an independent set. Folds and Hammer first studied the split graphs and were also introduced independently by Tishkovich and Cherniak. The split graph may have more than one click-through and stand-alone set; For example, the ABC path is a split graph whose vertices can be separated in three different ways:

Their forbidden induced subgraphs can characterize split graphs: An arbitrary split graph is if and only if no induced subgraph has four or five vertices or has a separate pair of edges (supplemented by a 4-round), Not far away. [4]

Critical factor graph

The Factor-critical graph is a graph with n vertices such that each sub-graph of the n-1 member has a full matching property. (Full matching in a graph means that a subset of edges. These are graphs in which each vertex subset is exactly the endpoint of one of the edges. Another type of connection in which all but one of the vertices is covered is called near-perfect matching. Similarly, a factor-crisis graph is a graph in which perfect close matches are omitted at each of the vertices.

Each round is a factor-critical graph along the person’s length, just as each complete graph with the number of vertices is a factor-critical graph. In general, any Hamiltonian graph with several vertices is a factor-critical graph. Friendship graphs (graphs created by connecting a set of triangles common to a vertex) are another example of a factor-critical graph, except that these graphs are non-Hamiltonian. Each claw-free connecting graph with the number of vertices is a factor-critical graph. For example, the 11-vertex graph, which removes a vertex from a regular bisector and is also connected, is factor-critical. This result is derived directly from a more fundamental theorem, in which every free-pawn graph with several vertex pairs must have a perfect match.

Graph Kelly

In mathematics, a Kelly graph is a graph that contains the algebraic structure of an algebraic group. Graph Kelly was proposed after Kelly’s theorem, which defines a group with a set of elements (usually the generator of that group).

Flat graph

In graph theory, a flat graph is a graph that can be enclosed in a plane. For example, a flat graph can draw so that its edges intersect only at the vertices. A non-planar graph is a graph that cannot plot so that its edges do not intersect at points other than the vertices. A flat graph drawn on the page without intersecting the edges is called a graph page or enclosed in a page.

A graph plane can be defined as a flat graph that describes each node as a point in two-dimensional space, and each edge as a curve in the plane such that the endpoints of each curve are points written from the nodes and the curves have nothing in common with each other. I do not have except at the endpoints. It is easy to see that the graph that can draw on the page can also be drawn in Korea.

Free Claw Graphs

In graph theory, which is part of mathematics, a free claw graph is a graph that has no claw subgraph. The claw is another name for the full two-part graph K1,3 (a star graph with three leaves and a central vertex). A free claw graph is a graph that has none of its sub-graphs; That is, each subgraph of the four vertices of that edge has more than three edges that connect them in this way. In other words, a free toe graph is a graph in which the complement of the inductive graph of the neighbors of each vertex of that graph is a free triangle. Free toe graphs were originally known as generalized edge graphs but became more important with the discovery of these three important properties:

Prove that every free toe graph of a vertex pair is a perfect match.

Discover polynomial solutions to find the maximum independent set in free paw graphs.

Describe the properties of the ideal free toe graph.

Show graphs

Different data structures are used to represent graphs in practice:

Proximity list: Rows are stored as records or objects, and each vertex stores a list of adjacent vertices. This data structure allows additional data to be stored at the top. Additional data can store if the edges are also stored as objects, in which case each vertex stores the incident edges, and each edge stores its acute vertex.

Adjacency Matrix: A two-dimensional matrix in which the rows represent the origin of the columns and the columns represent the vertices of the destination. Data at the edges and vertices must store outside the matrix. Only one edge can store between each pair of vertices.

Occurrence matrix: A two-dimensional Boolean matrix, in which rows represent vertices and columns represent edges. Inputs indicate whether the vertex of a row is connected to the edge of a column.