Relationship Graph Construction Example

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Task:

Construct a graph of the following relationship:

 

Define its properties.

Solution:

Construct graph G( X ) with the set of vertices  , and two vertices X i and X j are connected by an edge if and only if  . Since the relation    is symmetric, graph G( X ) is undirected.

Let’s construct the adjacency matrix (vertices) A:

X1 X 2 X 3 X 4 X 5 X 6
X1 1 1 1 1 1 1
X 2 1 1 1 1 1 0
X 3 1 1 1 1 0 0
X 4 1 1 1 0 0 0
X 5 1 0 0 0 0 0
X 6 1 0 0 0 0 0

Here Aij element is the number of edges going from vertex  X i   to vertex X j . As our graph is undirected, the adjacency matrix is symmetric.

Let’s construct the incidence matrix (ribs) R:

 

g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12
x1 1 1 1 1 1 0 0 0 0 1 0 0
x2 0 1 0 0 0 0 0 1 1 0 1 0
x3 0 1 0 0 0 0 0 1 1 0 0 1
x4 0 0 1 0 0 0 1 0 1 0 0 0
x5 0 0 0 1 0 1 0 0 0 0 0 0
x6 0 0 0 0 1 0 0 0 0 0 0 0

Here, the element Rij is 1 if the top vertex of the X i incident to edge g j and 0 otherwise.

Let’s construct the distance matrix D:

x1 x2 x3 x4 x5 x6
x1 0 1 1 1 1 1
x2 1 0 1 1 1 2
x3 1 1 0 1 2 2
x4 1 1 1 0 2 2
x5 0 0 0 1 0 1
x6 0 0 0 0 1 0

Here, the element Dij is the length the shortest path from vertex X i to vertex X j .

Since our graph is undirected, the distance matrix is symmetric.

We find the distance vector d, each of which is defined as the component
(the maximum distance from the vertex X i to every other vertex).

Distance vector d = (1, 2, 2, 2, 2, 2). Center is the peak vertex X1 , since it corresponds to the smallest distance (1 = d1 < d j , j = 2,…,6) .

Peripheral tops: X 2 , X 3 , X 4 , X 5 , X 6 , since it corresponds to the maximum remoteness (d j = 2, j = 2,…,6) .

Graph radius G( X ) is the center distance, r(G) = 1 .

Graph diameter G( X ) is the removal of the peripheral vertex, Diam(G) = 2 .

Let’s find the number for the internal and external stability graph. The largest set of internal stability for our graph has the form S = {X 4 , X 5 , X 6 } (the addition of any other peaks will receive adjacent vertices). Accordingly, graph G( X ) number is equal to the internal stability card(T ) = 1.

The smallest set for external stability for our graph is T X1 (as any other vertex (not owned to T) is connected to the apex of X1 from T). The number for the external sustainability graph G( X ) is equal to card(T ) 1.

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