Graph G is isomorphic to the complete graph K_n Prove that this statement is equ

Question

Graph G is isomorphic to the complete graph K_n
Prove that this statement is equivalent to using properties of anEulerian graph: G is connection, the number of verticies of g is n, and degree ofeach vertex=n-1 Graph G is isomorphic to the complete graph K_n
Prove that this statement is equivalent to using properties of anEulerian graph: G is connection, the number of verticies of g is n, and degree ofeach vertex=n-1
Prove that this statement is equivalent to using properties of anEulerian graph: G is connection, the number of verticies of g is n, and degree ofeach vertex=n-1 Graph G is isomorphic to the complete graph K_n
Prove that this statement is equivalent to using properties of anEulerian graph: G is connection, the number of verticies of g is n, and degree ofeach vertex=n-1

Explanation / Answer

Kn is a complete graph with n vertices. Thenany two vertices in Kn are adjacent. Suppose G is a graph isomorphic toKn ,  Then by definition G hasn vertices.      Also any two vertices of Gare adjacent.       So clearly G is a connected graph.           Each vertex of G is adjacent with all other vertices, that is n-1vertices, of G. Hence the degree of each vertex is n-1.     Convesely assume that G is connected,containing n vertices and degree of each vertex is n-1. That is each vertex is adjacent with n-1 vertices of G. Butthey are only n vertices of G. Hence each vertex is adjacent with all other vertices ofG. This shows that any two vertices of G is connected and hence Gis isomorphic to Kn.

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