Certainly! I can provide you with example codes in both MATLAB and Python for computing domination parameters in graph theory. Here are the codes for three commonly used domination parameters: domination number, total domination number, and connected domination number.

MATLAB code:

function domNumber = dominationNumber(adjacencyMatrix)

n = size(adjacencyMatrix, 1);

domNumber = Inf;

for i = 1:n

combinations = nchoosek(1:n, i);

numCombinations = size(combinations, 1);

for j = 1:numCombinations

domSet = combinations(j, :);

dominatedVertices = unique([domSet, find(any(adjacencyMatrix(:, domSet), 2))']);

if numel(dominatedVertices) == n

domNumber = min(domNumber, numel(domSet));

end

end

end

end

Python code:

import itertools

def domination_number(adjacency_matrix):

n = adjacency_matrix.shape[0]

dom_number = float('inf')

for i in range(1, n + 1):

combinations = itertools.combinations(range(n), i)

for dom_set in combinations:

dominated_vertices = set(dom_set)

for v in dom_set:

dominated_vertices |= set(np.nonzero(adjacency_matrix[v])[0])

if len(dominated_vertices) == n:

dom_number = min(dom_number, len(dom_set))

return dom_number

MATLAB code:

function totalDomNumber = totalDominationNumber(adjacencyMatrix)

n = size(adjacencyMatrix, 1);

totalDomNumber = Inf;

for i = 1:n

combinations = nchoosek(1:n, i);

numCombinations = size(combinations, 1);

for j = 1:numCombinations

domSet = combinations(j, :);

dominatedVertices = unique([domSet, find(any(adjacencyMatrix(:, domSet), 2))']);

if numel(dominatedVertices) == n && numel(unique(domSet)) == n

totalDomNumber = min(totalDomNumber, numel(domSet));

end

end

end

end

Python code:

import itertools

def total_domination_number(adjacency_matrix):

n = adjacency_matrix.shape[0]

total_dom_number = float('inf')

for i in range(1, n + 1):

combinations = itertools.combinations(range(n), i)

for dom_set in combinations:

dominated_vertices = set(dom_set)

for v in dom_set:

dominated_vertices |= set(np.nonzero(adjacency_matrix[v])[0])

if len(dominated_vertices) == n and len(set(dom_set)) == n:

total_dom_number = min(total_dom_number, len(dom_set))

return total_dom_number

MATLAB code:

function connDomNumber = connectedDominationNumber(adjacencyMatrix)

n = size(adjacencyMatrix, 1);

connDomNumber = Inf;

for i = 1:n

combinations = nchoosek(1:n, i);

numCombinations = size(combinations, 1);

for j = 1:numCombinations

domSet = combinations(j, :);

dominatedVertices = unique([domSet, find(any(adjacencyMatrix(:, domSet), 2))']);

subgraph = adjacencyMatrix(dominatedVertices, dominatedVertices);

if numel(dominatedVertices) == n && all(all(subgraph))

connDomNumber = min(connDomNumber, numel(domSet));

end

end

end

end

Python code:

import itertools

def connected_domination_number(adjacency_matrix):

n = adjacency_matrix.shape[0]

conn_dom_number = float('inf')

for i in range(1, n + 1):

combinations = itertools.combinations(range(n), i)

for dom_set in combinations:

dominated_vertices = set(dom_set)

for v in dom_set:

dominated_vertices |= set(np.nonzero(adjacency_matrix[v])[0])

subgraph = adjacency_matrix[list(dominated_vertices), :][:, list(dominated_vertices)]

if len(dominated_vertices) == n and np.all(subgraph):

conn_dom_number = min(conn_dom_number, len(dom_set))

return conn_dom_number

These codes calculate the domination number, total domination number, and connected domination number of a given graph using an exhaustive search. However, please note that for larger graphs, these brute-force algorithms may not be efficient, and more specialized algorithms might be required.

Meshgrid will be necessary to visualize the data.

I would also recommend using the optimization toolbox in Matlab.

In graph theory, dominance is a fundamental concept that measures how well a node or set of nodes controls or influences other nodes in the graph. There are several dominance parameters that can be computed in graph theory, including degree centrality, betweenness centrality, eigenvector centrality, and PageRank.

Here are some examples of MATLAB and Python codes that can be used to compute these dominance parameters:

1. Degree Centrality:

MATLAB code:

```

A = [0 1 1 0; 1 0 1 0; 1 1 0 1; 0 0 1 0]; % Adjacency matrix

D = sum(A, 2); % Degree matrix

C = D / (length(D) - 1); % Degree centrality vector

```

Python code:

```

import networkx as nx

G = nx.Graph([(0,1),(0,2),(1,2),(2,3)]) # Define graph

C = nx.degree_centrality(G) # Degree centrality dictionary

```

2. Betweenness Centrality:

MATLAB code:

```

A = [0 1 1 0; 1 0 1 0; 1 1 0 1; 0 0 1 0]; % Adjacency matrix

B = betweenness_centrality(sparse(A)); % Betweenness centrality vector

```

Python code:

```

import networkx as nx

G = nx.Graph([(0,1),(0,2),(1,2),(2,3)]) # Define graph

B = nx.betweenness_centrality(G) # Betweenness centrality dictionary

```

3. Eigenvector Centrality:

MATLAB code:

```

A = [0 1 1 0; 1 0 1 0; 1 1 0 1; 0 0 1 0]; % Adjacency matrix

[V, D] = eig(A); % Eigenvectors and eigenvalues

C = V(:,end) / sum(V(:,end)); % Eigenvector centrality vector

```

Python code:

```

import networkx as nx

G = nx.Graph([(0,1),(0,2),(1,2),(2,3)]) # Define graph

C = nx.eigenvector_centrality(G) # Eigenvector centrality dictionary

```

4. PageRank:

MATLAB code:

```

A = [0 1 1 0; 1 0 1 0; 1 1 0 1; 0 0 1 0]; % Adjacency matrix

P = sparse(A)'; % Transpose of adjacency matrix

alpha = 0.85; % Damping factor

C = pagerank(P, alpha); % PageRank vector

```

Python code:

```

import networkx as nx

G = nx.Graph([(0,1),(0,2),(1,2),(2,3)]) # Define graph

alpha = 0.85 # Damping factor

C = nx.pagerank(G, alpha) # PageRank dictionary

```

These codes provide examples of how to compute different dominance parameters in graph theory using MATLAB and Python. However, the specific code you use will depend on the particular graph you are analyzing and the dominance parameter you are interested in computing.

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