An Algorithmic-Modeling Approach to the Classification of Network Structures Using Boolean Algebra

Graph Theory and Algorithms: Application in Net- work Classification

Determining the Shortest Path in a Graph

To find the shortest path in a graph, the Ford algorithm is used. Consider a graph G with vertices x₁, x₂, . . . , x_n. Edges are defined as pairs (x_i, x_j) with weights l((x_i, x_j)). Each vertex x_i is assigned a value λ_i.

The algorithm proceeds as follows:

• Set λ₁ = 0, and for all other vertices λ_i = ∞,
• Iterate through all edges and perform relaxation: if λ_j > λ_i + l((x_i, x_j)), then set λ_j = λ_i + l((x_i, x_j)),
• Repeat until no changes occur.

After the algorithm completes, the values λ_i represent the shortest distances from vertex x₁ to all other vertices.

Figure 1. Application of Ford’s algorithm to a directed graph. The graph displays various distances between vertices with given weights.

Figure 1. Application of Ford’s algorithm to a directed graph. The graph displays various distances between vertices with given weights.

Adjacency Matrices and Permutations

The adjacency matrix of a graph represents the connections between its vertices. A permutation matrix P is used to prove isomorphism via the relation:

A₁ = P ⁻¹A₂P

Figure 2. Example of different adjacency matrices for isomorphic graphs. The structure is preserved under permutation.

Figure 2. Example of different adjacency matrices for isomorphic graphs. The structure is preserved under permutation.

Self-Complementary Graphs and Theorem

A graph G is self-complementary if it is isomorphic to its complement G. The following theorem holds:

Theorem: 

A self-complementary graph G has 4r or 4r + 1 vertices, where r is a non-negative integer.

Figure 3. Examples of self-complementary graphs. These graphs regain isomorphic struc- ture after adding the missing edges.

Figure 3. Examples of self-complementary graphs. These graphs regain isomorphic struc- ture after adding the missing edges.

Graph Operations: Union, Intersection, and Product

For arbitrary graphs G₁ = (X₁, U₁) and G₂ = (X₂, U₂), the following can be defined:

• Union: G1 ∪ G2 (logical disjunction in Boolean algebra)
• Intersection: G1 ∩ G2 (logical conjunction)
• Cartesian product: G1 × G2

Mathematically:

In Boolean algebra:

G₁ ∪ G₂ = OR(G₁, G₂),     G₁ ∩ G₂ = AND(G₁, G₂)

Figure 4. Examples of operations: union, intersection, and Cartesian product of two simple graphs G₁ and G₂. The first image shows input graphs, and the following illustrate the results of merging and pairing.

Figure 4. Examples of operations: union, intersection, and Cartesian product of two simple graphs G₁ and G₂. The first image shows input graphs, and the following illustrate the results of merging and pairing.

Trees and Their Relation to Boolean Algebra

A tree is a connected graph without cycles. Basic properties of trees include:

• A tree with n vertices has exactly n − 1 edges.
• Every connected acyclic graph is a tree.
• If the number of vertices is n and the number of edges is n − 1, the graph is a tree.

Boolean Algebra and Trees: Viewing each graph as a set of edges:

• The union of multiple trees forms a forest (disjoint set of trees).
• The intersection of trees yields common subgraphs that are also acyclic.

Figure 5. Different forms of trees – all illustrated graphs are connected and acyclic. They demonstrate possible structures formed by union or intersection of trees.

Figure 5. Different forms of trees – all illustrated graphs are connected and acyclic. They demonstrate possible structures formed by union or intersection of trees.

Planar Graphs and Euler’s Formula

Planar graphs are those that can be drawn in a plane without crossing edges.

Euler’s formula:

m − n + f = 2

where m is the number of edges, n the number of vertices, and f the number of faces.

Theorem: A graph is planar if it does not contain a subgraph isomorphic to K₅ or K_3,3, or their subdivisions.

Figure 6. Non-planar graphs: complete graph K₅ and complete bipartite graph K_3,3. These cannot be drawn on a plane without edge crossings.

Figure 6. Non-planar graphs: complete graph K₅ and complete bipartite graph K_3,3. These cannot be drawn on a plane without edge crossings.

According to Kuratowski’s theorem, the planarity of a graph can be determined only if it does not contain a subgraph isomorphic to K₅ or K_3,3 or their subdivisions. This characterization is crucial for graph classification in discrete mathematics and computer science.

Networks, Semilattices and the Application of Boolean Algebra in Classification

Mathematical Approach to Graphs, Intersections and Boolean Proofs for G_k, U_k, H_k, B_k

G_k – Generalized Networks (Weak Relations)

Let the relations be R₁ = {(a, a), (a, b), (b, c)} and R₂ = {(b, b), (b, c), (c, c)}.

Union: R₁ ∪ R₂ = {(a, a), (a, b), (b, c), (b, b), (c, c)}

Intersection:R₁ ∩ R₂ = {(b, c)}

Mathematical interpretation: Via line graphs:

If P₁ : y = x (reflexivity), P₂ : y = x + 1 (transitive relation), their intersection is found by solving:

⇒ no solution ⇒ no common points

Boolean table:

a	b	a ∨ b
0	0	0
0	1	1
1	0	1
1	1	1

Conclusion: G_k structures have partial Boolean compatibility but lack complements and boundaries.

U_k – Union-Based Networks

Let A = {1, 2} and B = {2, 3}.

Union: A ∪ B = {1, 2, 3}

Intersection:A ∩ B = {2} (not used in U_k)

Line equations: Connect points in plane: (1, 0) and (2, 0) form y = 0 (horizontal) (2, 0) and (2, 1) form x = 2 (vertical).

Intersection:x = 2, y = 0 ⇒ (2, 0) — point of intersection Boolean analysis: Only ∨ is defined: 1 ∨ 0 = 1, 1 ∨ 1 = 1 Conclusion: U_k represents join logic without shared base (no meet).

H_k – Horizontal-Vertical Networks

Let A = {(1, 0), (2, 0)}, B = {(2, 0), (2, 1)}.

Union:A ∪ B = {(1, 0), (2, 0), (2, 1)}

Intersection:A ∩ B = {(2, 0)}

Mathematical proof: Line y = 0 (horizontal) and x = 2 (vertical) intersect at (2, 0).

Boolean form: ∨ = union = all edges, ∧ = intersection = common node

Table:

a	b	a ∨ b	a ∧ b
0	0	0	0
1	0	1	0
0	1	1	0
1	1	1	1

Conclusion: H_k networks allow multidimensional interpretation of relations through lines and nodes.

B_k – Complete Boolean Networks

A = {0, 1}, B = {1, 2}, universal set U = {0, 1, 2, 3}

Operations:

A ∪ B = {0, 1, 2}, A ∩ B = {1}, A^′ = {2, 3}, B^′ = {0, 3}

Boolean laws:

A ∨ B = A ∪ B,     A ∧ B = A ∩ B

A ∨ A^′ = U,     A ∧ A^′ = ∅

Line equations: If elements are represented as points in coordinate plane:

• A: x = 0, x = 1 B: x = 1, x = 2
• Line x = 1 — common intersection, gives node (1, y) for any y

Truth table:

a	b	a ∨ b	a ∧ b	¬a
0	0	0	0	1
0	1	1	0	1
1	0	1	0	0
1	1	1	1	0

Conclusion: B_k networks satisfy all logical operations and serve as a complete base for system implementation using Boolean algebra.

Networks, Semilattices, and the Application of Boolean Algebra in Classification

Optimization Algorithm for Classification and Mapping of Net- work Structures using Boolean Algebra

To enhance the efficiency of network structure classification, we developed an algorithm that applies minimization of Boolean expressions in Disjunctive Normal Form (DNF) and Conjunctive Normal Form (KNF). This approach simplifies the structure of directions, intersections, and unions within the network.

Algorithm Steps:

• Input: A set of relations between network nodes expressed as binary expressions (e.g., x₁ ∨ x₂, x₁ ∧ x₃).
• Transformation: Conversion of expressions into DNF and KNF formats.
• Minimization: Use of methods such as the Quine–McCluskey algorithm or Kar- naugh maps to minimize logical expressions.
• Direction Analysis: Identification of line equations of the form y = ax + b for each relation.
• Intersections: Calculation of intersection points by solving systems of linear equa- tions for each pair of relations.
• Output: An optimized set of network connections, classified according to the Gk, Uk, Hk, Bk typology.

Example:

Let us consider the relations:

R₁ = x₁ ∨ x₂, R₂ = x₂ ∧ x₃

We transform them as follows:

DNF: f (x₁, x₂, x₃) = (x₁ ∧ ¬x₂) ∨ (x₂ ∧ x₃)

KNF: f (x₁, x₂, x₃) = (x₁ ∨ x₂) ∧ (¬x₁ ∨ x₃)

The intersection of the lines y = x₁ and y = −x₂ +3 is obtained by solving the system:

Conclusion: By optimizing expressions and accurately calculating intersections and unions, we enable improved interpretation and automation of network classification with implementation in MATLAB or Python.

Boolean Algebra as a Lattice Extension

Boolean algebra introduces additional requirements:

• distributivity of ∨ and ∧ operations,
• existence of a complement a for every element a,
• existence of bounds: 0 (least) and 1 (greatest element).

Examples of identities in Boolean algebra:

Truth Table and Logical Interpretation

For elements a, b ∈ {0, 1}:

a	b	a ∨ b	a ∧ b
0	0	0	0
0	1	1	0
1	0	1	0
1	1	1	1

This table shows the basic logical operations of union and intersection within Boolean algebra.

Operational Interpretation for G_k–B_k Structures in Boolean Al- gebra

-

U_k (union networks): Closed only under ∨ (symbol 0), while ∧ (1) is undefined.

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T_k (topological networks): Define localized directions in networks with partial connections. Partial ∧ operations are allowed:

-

H_k (horizontal-vertical networks):  Define clear two-dimensional structure. ∨ represents horizontal joins, and ∧ vertical intersections.

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B_k (Boolean networks): Satisfy all Boolean algebra axioms:

Mathematical Function and Proof via Boolean Algebra for Net- work Structures

We develop an advanced formal function for classifying networks and sub-networks via set union and intersection, mapped onto Boolean join (symbol 0) and meet (symbol 1) operations.

Formal Function: Let A, B ⊆ U be subsets of the universal set U , and define the function f as:

Mathematical Proof: Let U = {0, 1}² with elements:

(0, 0), (0, 1), (1, 0), (1, 1)

Let A = {(1, 0), (0, 1)} and B = {(0, 1), (1, 1)}. Then:

A ∪ B = {(1, 0), (0, 1), (1, 1)},    A ∩ B = {(0, 1)}

Symbolically:

A ∨ B = logical disjunction,     A ∧ B = logical

conjunction Thus, f (A, B) = 1, confirming Boolean compatibility.

Axiomatic Properties:

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Commutativity: A ∨ B = B ∨ A, A ∧ B = B ∧ A

-

Associativity: A ∨ (B ∨ C) = (A ∨ B) ∨ C

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Distributivity: A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)

-

Complement:

Operations 0 and 1:

-

0: Join (union) – operator ∨

-

1: Meet (intersection) – operator ∧

Example with networks: If G₁ = (V₁, E₁) and G₂ = (V₂, E₂), then:

G₁ ∨ G₂ = (V₁ ∪ V₂, E₁ ∪ E₂), G₁ ∧

G₂ = (V₁ ∩ V₂, E₁ ∩ E₂)

Application of the function: Define the characteristic function:

Then:

χ_A∪B(x) = χ_A(x) ∨ χ_B(x),    χ_A∩B(x) = χ_A(x) ∧ χ_B(x)

Truth Table:

a	b	a ∨ b	a ∧ b	¬a
0	0	0	0	1
0	1	1	0	1
1	0	1	0	0
1	1	1	1	0

Disjunctive Normal Form (DNF):

f (a, b) = (¬a ∧ b) ∨ (a ∧ ¬b) ∨ (a ∧ b)

Conjunctive Normal Form (CNF):

f (a, b) = (a ∨ b) ∧ (¬a ∨ b) ∧ (a ∨ ¬b)

Minimal Form:

f (a, b) = a ∨ b

Conclusion: By linking set theory, logic, and graph theory via f (A, B) and truth tables, a formal framework is established for systematic classification of network structures using Boolean algebra. This enables automated testing and application in digital logic, lattice theory, and classification algorithms.

Application for Representing Networks and Subnetworks Using Boolean Algebra

Figure 1. Application Interface.

Figure 1. Application Interface.

Figure 2. Network Visualization.

Figure 2. Network Visualization.

Figure 3. Logical Functions and Their Graphical Representations in the Context of Boolean Algebra.

Figure 3. Logical Functions and Their Graphical Representations in the Context of Boolean Algebra.

• Logical expressions at the top of the figure:

  o

  The function f(x1,x2,x3)=x1∨x2∨x3 represents a disjunction (logical "OR") of three variables. It is an example of a simple logical function in disjunctive form.

  o

  The function g(x1,x2)=x1∧¬x1∧¬x2 is a contradictory function (never true), used to represent paradoxical or unachievable logical connections.

  o

  From these symbolic functions, relationships between network nodes can be constructed based on the truth values of the expressions.

• Truth table (bottom left):

  o

  Shows all possible combinations of values for variabl x1 and x2, along with the corresponding function result.

  o

  The table is used to generate DNF (Disjunctive Normal Form) and KNF (Conjunctive Normal Form), essential for logic minimization and further network analysis.

• Graphs (right side):

  o

  The upper graph shows a simple network based on the function ggg, connecting only nodes where the expression evaluates to true.

  o

  The lower graph represents a more complex network, possibly derived from combining multiple logical expressions (such as h(x2,x3)), where nodes interact based on logical evaluation.

Figure 4. Adjacency Matrices and Graphs for Gk, Bk, Tk and Other Networks.

Figure 4. Adjacency Matrices and Graphs for Gk, Bk, Tk and Other Networks.

• Top left – Gk (Generalized Network):

  o

  Displays a complex graph with multiple node connections.

  o

  The adjacency matrix indicates which nodes are connected (1 for connection, 0 for none).

  o

  This type illustrates a general, unstructured network without predefined logical restrictions.

• Top right – Bk (Boolean Network):

  o

  A network where relationships are defined using Boolean logic.

  o

  The adjacency matrix reflects logical rules through partial node connections.

  o

  Often used for modeling systems like logic processors and digital circuits.

• Bottom left – Tk (Topological Network):

  o

  Shows a tree structure with clearly defined hierarchy and directional edges (from root to leaves).

  o

  The matrix displays outgoing edges per node, with “2” indicating two child connections.

  o

  Useful in modeling organizational charts, inheritance trees, or search algorithms.

• Bottom right – Random Unstructured Network:

  o

  Visualized as a complex mesh of randomly interconnected nodes.

  o

  The adjacency matrix shows highly specific node connections.

  o

  This model is typical for neural networks or social networks with limited links.

Figure 5. Logical Functions, Truth Tables, and Related Networks.

Figure 5. Logical Functions, Truth Tables, and Related Networks.

• Top – Functions f(x),g(x)f(x), g(x)f(x),g(x):

  o

  f(x1,x2,x3)=x1∨x2∨x3 is a disjunction that outputs 1 if at least one input is 1.

  o

  o g(x1,x2)=x1∧¬x1∧¬x2 is inherently false (as x1∧¬x1 is always 0), thus always evaluates to 0.

• Middle left – Truth Table:

  o

  Shows all binary combinations for x1,,x2 and the result for f(x1,x2).

  o

  Used for analyzing and evaluating Boolean expressions.

  o

  Enables logic minimization into DNF and KNF.

• Right – Graphs Linked to Functions:

  o

  The upper graph shows a simple three-node structure, possibly representing g(x1,x2) or a binary tree.

  o

  The lower graph is more complex, likely illustrating h(x2,x3), involving more connections and interactions.

  o

  These graphs visually represent logical functions as network structures.

Figure 6. Boolean Algebra Axioms and Network Types: Uk, Gk, Tk, Bk, and Subnetworks.

Figure 6. Boolean Algebra Axioms and Network Types: Uk, Gk, Tk, Bk, and Subnetworks.

• Top left – Boolean Algebra Axioms:

  o

  Presents fundamental Boolean rules:

  ▪

  a∧0=0

  ▪

  a∧1=a

  ▪

  a∨0=a

  ▪

  a∧a=a

  ▪

  a∨a=a

• Middle and Right – Network Visualizations:

  o

  Uk (Union Network): One central node connected to several others producing output 1, representing disjunction.

  o

  Gk (Generalized Network): A complex web of nodes, possibly with loops or cycles.

  o

  Tk (Topological Network): Hierarchical with clear directionality (e.g., tree structures).

  o

  Bk (Boolean Network): Dense and interconnected, based on logical rules.

• Bottom – Subnetworks and Network Structure:

  o

  A subnetwork is a subset of a larger network preserving the same structure and rules.

  o

  These visualizations show how logic expressions, once translated into graphs, can be decomposed into smaller functional units.

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