Invariants are graph properties that depends only on the abstract structure, not on particular labeling or drawings of the graph. Examples include the number of vertices and the number of edges. Here we list all the graph operations and invariants already inserted in the project, with the respective library used for implementation.

They can be used in the writing of (in)equations or after filtering, to study the output graphs.

Structural
Chromatic number (χ)	Matching number (ν)	Radius (r)
Number of vertices (n)	Number of components (w)	Vertex connectivity (ϰ)
Number of edges (Е)	Degree regularity (dᵣ)	Edge connectivity (λ)
Clique number (ω)	Maximum degree (Δ)	Minimum edge cover number (mec)
Independence number (⍺)	Minimum degree (ẟ)	Number of triangles (Τ)
Total domination number (Ɣₜ)	Average degree (dₐ)	Wiener Index (W)
Domination number (Ɣ)	Vertex cover number (τ)	Number of Spanning trees (ʈ)
Girth (ɡ)	Diameter (diam)	Density (Ɗ)

Spectral
Largest M-eigenvalue M=A, L, Q, N, S, D, DL, DQ, E	2th Largest M-eigenvalue M=A, L, Q, N, S, D, DL, DQ, E	Smallest M-Eigenvalue M=A, Q, N, S, D, DL, DQ, E
M-Energy M=A, L, Q, N, S, D, DL, DQ, E	Algebraic connectivity (ac)	Estrada Index (EE)
Nullity (η)	Number main M-eigenvalue M=A, D, Q, S	Rank M- matrix M=A, L, Q, D, N
Determinant M M=A, L, Q, N, S, D, DL, DQ, E

The definition of M matrices can be found in Dictionary Matrix M.

To build the in(equations) the user can calculate the numeric invariants not only in the graph, but can also use the line graph and complement

They can be used in the conditions or after filtering, to study the output graphs.

Structural
Planar	Regular	Connected	Triangle-free
Biconnected	Tree	Bipartite	Bull-free
Self-complementary	Eulerian	Regular Transmission	Claw-free
Chordal	Has bridge	Is threshold	Cubic

Spectral *
Some M-eigenvalue integer M=A, L, Q, S, N, D, DL or DQ	M-integral M=A, L, Q, N, S, D, DL, DQ, E
Largest M-eigenvalue is integer M=A, L, Q, N, S, D, DL, DQ, E	Matrix M is invertible M=A, L, Q, N, S, D, E

They are invariants that are neither numeric nor boolean, but can be used to study the graphs returned in the filtering.

Structural
Dominating Set	Eigenvector Centrality
Maximum Clique	Closeness Centrality
Minimum edge cover (set)	Betweeness Centrality
Maximum Matching Set	Harmonic Centrality
Maximum Independent Set	Transmission

Spectral *
Spectrum M M= A, L, Q, N, D, S, DL, DQ, E	M-Matrix M= A, I, L, Q, N, D, S, DL, DQ, E
Eigenvectors M= A, L, Q, N, D, S, DL, DQ, E

You can also use several usual functions in mathematics:

Trigonometric	Basic	Functions	Utils	Logic	Constants	Symbols
Sine	Sum	Natural Logarithm	Floor	AND	PI	Equal
Cosine	Subtraction	Logarithm	Ceiling	OR		Different
Tangent	Division	Square Root	Modulus			Less Than
	Product	Modulus				Greater Than
	Power					Less or Equal Than
						Greater or Equal Than

These are basic operations that are applied to a single graph and return another graph.

Graph Operations
Complementary Graph	Line Graph
Clique Maximal	Graph

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Where the matrix M can be:

• A: Adjacency
• L: Laplacian
• Q: Signless Laplacian
• N: Normalized Laplacian
• D: Distance
• DL: Distance Laplacian
• DQ: Distance Signless Laplacian
• S: Seidel
• E: Eccentricity

The definitions is available on Dictionary Matrix M.