This page aims to help newcomers to understand the invariants and graphs options present in Graph Filter.

Implementation by Networkx Python library

It's the smallest number of colors needed to color the vertices of G so that no two adjacent vertices share the same color.

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Implementation by Networkx Python library

It's the smallest number of colors needed to color the edges of G so that no two adjacent edges share the same color.

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Implementation by Networkx Python library

It's the quantity of vertices present on the graph.

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Implementation by Networkx Python library

It's the quantity of edges present on the graph.

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Implementation by Grinpy and Netowrokx Python library

A clique is a subset of nodes in which each pair of nodes is adjacent. The clique number of a graph G is the number of vertices in a maximum clique in G.

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Implementation by Grinpy Python library

Independent set or stable set is a set of vertices in a graph, no two of which are adjacent. The independence number of a graph G is the number of vertices in a maximum independent set in G.

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Implementation by Grinpy Python library

A dominating set for a graph G=(V, E) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number is the number of vertices in a smallest dominating set for G.

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Implementation by Grinpy Python library

The total domination number of a graph is the size of a smallest total dominating set, where a total dominating set is a set of vertices of the graph such that all vertices (including those in the set itself) have a neighbor in the set. Total dominating numbers are defined only for graphs having no isolated vertex

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Implementation by Grinpy Python library

The connected domination number of a connected graph G, denoted d(G), is the size of a minimum connected dominating set of a graph G. A connected dominating set in a connected graph G is a dominating set in G whose vertices induce a connected subgraph

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Implementation by Grinpy Python library

A matching in a graph is a set of edges in which no two distinct edges share a common endpoint. The matching number of graph G, or edge independence number, is the size of a maximum independent edge set.

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Implementation by Grinpy Python library

It's the minimum number of nodes that must be removed to disconnect the graph or render it trivial.

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Implementation by Grinpy Python library

It's the minimum number of edges that must be removed to disconnect the graph.

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Implementation by NetworkX Python Library

It's the number of maximals componnents in the graph

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Implementation by

It's the number of distinct values in the sequence of degrees of the graph.

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Implementation by NetworkX Python Library

It's the largest vertex degree of G

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Implementation by NetworkX Python Library

It's the smallest vertex degree of G

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Implementation by NetworkX Python Library

It's the average degree per vertex in the graph

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Implementation by NetworkX Python Library

It's the length of a shortest cycle contained in the graph.

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Implementation by NetworkX Python Library

An edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. The minimum edge cover (set) is an edge covering of smallest cardinality.

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Implementation by Grinpy Python library

It's the size of a minimum vertex cover. A vertex cover (or node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph.

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Implementation by NetworkX Python Library

It's the greatest distance between any pair of vertices.

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Implementation by NetworkX Python Library

The radius r of a graph is the minimum eccentricity of any vertex. The eccentricity of a vertex is the greatest distance between it and any other vertex.

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Implementation by Networkx Python library. The matrix implemented on software are:

• Adjacency Matrix - A: It's a square (0,1)-matrix whose elements indicate whether pairs of vertices are adjacent, with the value 1, or not, with the value 0.
• Laplacian - L: It's L=D-A, where D is the diagonal matrix of node degrees and A is the adjacency matrix of the graph.
• Signless Laplacian Matrix - Q: It's L=D+A, where D is the diagonal matrix of node degrees and A is the adjacency matrix of the graph.
• Normalized Laplacian Matrix - N: It's the matrix N=D^½LD^½, where L is the Laplacian matrix and D is the diagonal matrix of node degrees.
• Incidence Matrix: It's a (0,1)-matrix which has a row for each vertex and column for each edge, and an entry (v,e)=1 iff vertex v is incident upon edge e.
• Distance Matrix - D: It's square matrix containing the distances, taken pairwise, between the vertices.
• Seidel Matrix - S: It's square matrix S=J-I-2A, where J is the matrix formed by 1, A is the adjacency matrix and I the identity matrix
• Laplacian Distance Matrix - DL: It's square matrix DL=Diag(Tr)-D, where Diag(Tr) is the diagonal matrix of the vertex transmissions and D is the distance matrix.
• Signless Laplacian Distance Matrix - DQ: It's square matrix DL=Diag(Tr)+D, where Diag(Tr) is the diagonal matrix of the vertex transmissions and D is the distance matrix.
• Eccentricity matrix - E: It's square matrix constructed from the distance matrix, retaining the largest distances in each row and each column, while other elements of the distance matrix are set to zero.

• Visualization and Filtering (Matrix, Spectrum, Main eigenvalues, Eigenvectores, Energy or Determinant)

Implementation by Numpy Python library

It's the largest eigenvalue of matrix M.

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Implementation by NetworkX Python Library

It's the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. This is L=D-A, where D is the degree matrix and A is the adjacency matrix of the graph.

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Implementation by Numpy Python Library

Energy of a graph is E= Σ |λ - Tr(M)/n|, where Σ are the eigenvalues of M, Tr(M) is the trace of matrix and n the order.

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Implementation by own implementation

Checks if there is an integer M-eigenvalue

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Implementation by Numpy Python Library

It's the set of eigenvalues of matrix M

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Implementation by Numpy Python Library

It is the set of eigenvectors and their eigenvalues in the matrix M

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Implementation by own implementation

Checks if all M-eigenvalues are integer.

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Implementation by Numpy Python Library

This corresponds to the maximal number of linearly independent columns of matrix M.

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Implementation by Numpy Python Library

Check if matrix M is invertible.

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Implementation by Numpy Python Library

Calculates the determinant of matrix M.

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Implementation by own implementation

Checks whether the largest M-eigenvalue is an integer.

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Implementation by Numpy and own implementation

The eigenvalue x of a matrix M is said to be a main eigenvalue of G, if the eigenspace E(x) is not orthogonal to the all-1 vector.

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Implementation by NetworkX Python Library

The Wiener index (or the "path number") is a graph index defined for a graph on n nodes by W=½ΣΣ d(i,j), where d(i,j) is the distance between the vertices i and j.

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Implementation by NetworkX Python Library

It's a graph index defined for a graph on n nodes by EE=Σ e^λ, where λ are the eigenvalues of adjacency matrix.

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Implementation by NetworkX Python Library

In the matrix theory of graphs, the nullity of the graph is the nullity of the adjacency matrix A of the graph. This nullity equals the multiplicity of the eigenvalue 0 in the spectrum of the adjacency matrix

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Implementation by Numpy Python Library

The number of spanning trees of a connected graph G. A spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. This invariant can be obtained from Kirchhoff's theorem.

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Implementation by NetworkX Python Library

The graph density of simple graphs is defined to be the ratio of the number of edges with respect to the maximum possible edges The density is 0 for a graph without edges and 1 for a complete graph.

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Implementation by NetworkX Python Library

It's 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.

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Implementation by NetworkX Python Library

There is a path from any point to any other point in the graph.

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Implementation by NetworkX Python Library

A graph is biconnected if, and only if, it cannot be disconnected by removing only one node (and all edges incident on that node).

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Implementation by NetworkX Python Library

It's a graph whose vertices can be divided into two disjoint and independent sets U and V such that every edge connects a vertex in U to one in V.

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Implementation by NetworkX Python Library

A bridge is an edge of a graph whose deletion increases the graph's number of connected components.

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Implementation by NetworkX Python Library

It's a graph that has an Eulerian cycle. An Eulerian cycle (Eulerian circuit or Euler tour) in an undirected graph is a cycle that uses each edge exactly once..

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Implementation by Grinpy Python library

It's one in which all induced cycle in the graph should have exactly three vertices.

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Implementation by Networkx Python library

Is a graph that can be constructed from a one-vertex graph by repeated applications of the following two operations: addition of a single isolated vertex or a single dominating vertex, to the graph.

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Implementation by Grinpy Python library

It's a graph in which no three vertices form a triangle of edges.

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Implementation by Grinpy Python library

The bull graph is a triangle with two disjoint pendant edges.

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Implementation by Grinpy Python library

It's a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree

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Implementation by Grinpy Python library

It's a regular graph with vertices of degree 3.

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Implementation by Grinpy Python library

For each vertex, it calculates the sum of its distances from the other vertices. A graph is Regular Transmission if all sums has the same value.

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Implementation by Networkx Python library

A self-complementary graph is a graph which is isomorphic to its complement.

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Implementation by NetworkX Python Library

The complement of a graph G is a graph Comp(G) on the same vertices such that two distinct vertices of c(G) are adjacent if and only if they are not adjacent in G.

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Implementation by NetworkX Python Library

The line graph of an undirected graph G is another graph ℓ(G) that represents the adjacencies between edges of G. It's constructed in the following way: for each edge in G, make a vertex in ℓ(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in ℓ(G).

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Implementation by NetworkX Python Library

Returns the clique graph of the graph. The nodes of the clique graph Ƙ(G) are the maximal cliques of G and an edge joins two cliques if the cliques are not disjoint.

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Implementation by NetworkX Python Library

For each vertex its degree is calculated

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Implementation by NetworkX Python Library

For node i is the i-th element of the normalized eigenvector associated with the largest adjacency eigenvalue

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Implementation by NetworkX Python Library

For each node is the average length of the shortest path between the node and all other nodes in the graph.

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Implementation by NetworkX Python Library

For each nodes computes the number of times the node acts as a bridge along the shortest path between two other nodes.

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Implementation by NetworkX Python Library

For each vertex computes the sum of the inverses of the distances from this vertex to the rest of the graph is calculated.

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Implementation by NetworkX Python Library

An order-zero graph, which has no vertices

• None

Implementation by NetworkX Python Library

A graph from an ASCII sequence of characters

• A valid sequence of characters

Implementation by NetworkX Python Library

A graph consisting of a single closed loop, where each node is connected to exactly two other nodes in a circular fashion.

• Number of nodes greater than 0 and less than or equal to 60

Implementation by NetworkX Python Library

A graph that consists of a linear sequence of nodes, where each node is connected to its adjacent nodes in the sequence.

• Number of nodes greater than 0 and less than or equal to 60

Implementation by NetworkX Python Library

A graph where each pair of distinct nodes is connected by a unique edge, resulting in every node being directly connected to every other node in the graph.

• Number of nodes greater than 0 and less than or equal to 60

Implementation by NetworkX Python Library

A graph that consists of a central node connected to all other nodes in the graph, while the other nodes are not directly connected to each other.

• Number of nodes greater than 0 and less than or equal to 60

Implementation by NetworkX Python Library

A graph that has the maximum possible number of edges among all graphs with the same number of vertices and avoiding a given complete graph as a subgraph.

• Number of nodes greater than 0 and less than or equal to 60
• Number of subsets greater than 0 and less than or equal to number of nodes

Implementation by NetworkX Python Library

A graph structure that represents a grid-like arrangement of nodes. It consists of a set of nodes arranged in rows and columns, where each node is connected to its adjacent nodes horizontally and vertically.

• Number of rows multiplied by number of columns greater than 0 and less than or equal to 60

Implementation by NetworkX Python Library

A graph where nodes are arranged in a triangular pattern. It consists of equilateral triangles where each node is connected to its three neighboring nodes.

• Number of rows multiplied by number of columns greater than 0 and less than or equal to 60

Implementation by NetworkX Python Library

A non-planar graph with 10 vertices and 15 edges. It features a central pentagon and five additional edges connecting each pentagon vertex to an external vertex.

• None

Implementation by NetworkX Python Library

A graph where every vertex has an equal number of neighboring vertices, known as its degree.

• Degree of nodes greater than 0 and less than the number of nodes
• Degree of nodes multiplied by number of vertices must be even

Implementation by NetworkX Python Library

A graph constructed by recursively combining smaller graphs through disjoint union and complementation operations.

• K number greater than 0 and less than 6

The graph created will have a number of nodes equal to 2^k

Implementation by NetworkX Python Library

A graph consisting of two disjoint sets of vertices, where every vertex in one set is connected to every vertex in the other set.

• The sum between the number of nodes of the two sets greater than 0 and less than or equal to 60

Implementation by NetworkX Python Library

A graph where its vertices are defined by a binary sequence.

1. If the bit value is one, an universal vertex is added
2. If the value of the bit is zero, an isolated vertex is added

• A binary sequence of size greater than 0 and less than or equal to 60