How to find SingleNodeCycle and euler cycle cost of a complete graph

x = [1 1 1 2 2 3 ];

y = [ 2 3 4 3 4 4];

these are the co-ordinates of a graph, i have created a complete graph from these points and i wanted to find SingleNodeCycle coset and Euler cost of the graph.

Answer 1

[BhaTTa]

@Ashish Verma, to analyze a graph based on given coordinates and compute properties like Single Node Cycle cosets and Euler cost, we need to first construct the graph and then apply the relevant graph theory concepts. Here's how you can proceed using MATLAB:

Step 1: Construct the Graph

Given the coordinates, you can construct a graph using the graph function in MATLAB. The coordinates represent edges between nodes.

Step 2: Analyze the Graph

Euler Cost

The Euler cost in a graph context often relates to finding an Eulerian path or circuit, which is a path or circuit visiting every edge exactly once. For a graph to have an Eulerian circuit, every vertex must have an even degree. For an Eulerian path, exactly two vertices can have an odd degree.

Below is the example implementation:

x = [1 1 1 2 2 3];
y = [2 3 4 3 4 4];

% Create the graph
G = graph(x, y);% Check degrees of vertices
plot(G);

degrees = degree(G);

% Check for Eulerian circuit
hasEulerianCircuit = all(mod(degrees, 2) == 0);

% Check for Eulerian path
oddDegreeVertices = sum(mod(degrees, 2) == 1);
hasEulerianPath = oddDegreeVertices == 0 || oddDegreeVertices == 2;

% Display results
if hasEulerianCircuit
    disp('The graph has an Eulerian circuit.');
elseif hasEulerianPath
    disp('The graph has an Eulerian path.');
else
    disp('The graph does not have an Eulerian path or circuit.');
end