Answer to Question #195053 in Discrete Mathematics for vasker

Question #195053

. 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?


1
Expert's answer
2021-05-19T16:31:42-0400

"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=\\\\\n=\\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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