Solution to The complete m-partite graph π‘²π’πŸ,π’πŸ,……,π’π’Ž has vertices partitioned into m subsets of π’πŸ, π’πŸ, … - Sikademy
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The complete m-partite graph π‘²π’πŸ,π’πŸ,……,π’π’Ž has vertices partitioned into m subsets of π’πŸ, π’πŸ, … … , π’π’Ž elements each, and vertices are adjacent if and only if they are in different subsets in the partition. For example, if m=2, it is our ever-trusting friend, a complete bipartite graph. a. Draw these graphs: i. π‘²πŸ,𝟐,πŸ‘ ii. π‘²πŸ,𝟐,πŸ‘ iii. π‘²πŸ’,πŸ’,πŸ“,𝟏 b. How many vertices and how many edges does the complete π‘²π’πŸ,π’πŸ,……,π’π’Ž have?

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K_{1,2,3}


K_{2,2,3}


K_{4,4,5,1}

the degree of the top of the i-th share

\sum\limits_{i\neq j}mj

then

S=\sum mj=> \sum\limits_{i\neq j}mj=S-mi

the sum of the degrees of the vertex of the i-th part

(S-mi)*mi=Smi-mi^2


sum of degrees of all vertices

\sum Smi-mi^2=S\sum mi-\sum mi^2=S^2-\sum mi^2=(\sum mi)^2 -\sum mi^2=\\ =\sum\limits_{i<j}2mi*mj

the number of edges in the graph

\frac{1}{2}\sum\limits_{i<j}2mi*mj=\sum\limits_{i<j}mi*mj

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Question ID: mtid-5-stid-8-sqid-2982-qpid-1681