The construction of a regular graph
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1 The construction of a regular graph Ryosuke MIZUNO Department of Physics Kyoto University Yawara ISHIDA Research Institute for Mathematical Science Kyoto University
2 Graphs of Diameter=2 in Graph Golf (Order,Degree)=(64,16) (Order,Degree)=(256,23)
3 Outline (1) How to make a regular graph (2) Graphs of Diameter=2 in Graph Golf (3) Summary
4 How to make a regular graph (A) Making a regular graph directly (B) Making a regular graph from small regular graphs
5 Constructing methods (Diameter=2) (A) (B) Brown s Construction Generalized Brown s Construction Copying pentagon Copying G 8 Strong product Order Ex. 5k 8k n,m nm Degree k+1 k+2 d 1,d 2 d 1 d 2 +d 1 +d 2 Order/De gree^ /k 8/k (p : a prime k : a natural number )
6 Constructing methods (Diameter=2) (A) (B) Brown s Construction Generalized Brown s Construction Copying pentagon Copying G 8 Strong product Order Ex. 5k 8k n,m nm Degree k+1 k+2 d 1,d 2 d 1 d 2 +d 1 +d 2 Order/De gree^ /k 8/k (p : a prime k : a natural number )
7 Constructing method : Copying G 8 (top,0) (top,1) G 8 (middle,0) (middle,1) (middle,2) A regular graph of order=8, degree=3 and diameter=2 (bottom,0) (bottom,1) (bottom,2)
8 Constructing method : Copying G 8 Connect k-copies of G 8 as follows: (for all i j. i, j = 1, 2,, k ) ( i,(t,0)) ( i,(t,1)) ( i,(m,0)) ( i,(m,2)) ( i,(m,1)) ( i,(b,0)) ( i,(b,2)) ( i,(b,1)) ( j,(t,0)) ( j,(t,1)) ( j,(m,0)) ( j,(m,2)) ( i,(m,1)) ( j,(b,0)) ( j,(b,2)) ( j,(b,1)) 1 k This Graph has 8k nodes degree=k+2
9 Constructing methods (Diameter=2) (A) (B) Brown s Construction Generalized Brown s Construction Copying pentagon Copying G 8 Strong product Order Ex. 5k 8k n,m nm Degree k+1 k+2 d 1,d 2 d 1 d 2 +d 1 +d 2 Order/De gree^ /k 8/k (p : a prime k : a natural number )
10 Outline (1) How to make a regular graph (2) Graphs of Diameter=2 in Graph Golf (3) Summary
11 Graphs of Diameter=2 in Graph Golf (Order,Degree)=(64,16) (Order,Degree)=(256,23)
12 Graphs of Diameter=2 in Graph Golf (Order,Degree)=(64,16)
13 How to make the (64,16) graph Connecting 8-copies of G 8 Order=8 Degree=3 8 The graph of Order = 8*8 = 64 Degree = = 10
14 How to make the (64,16) graph Connecting 8-copies of G 8 G 8 8 Order=8 Degree=3 8 The graph of Order = 8*8 = 64 Degree = = 10
15 Graphs of Diameter=2 in Graph Golf (Order,Degree)=(64,16) (Order,Degree)=(256,23)
16 Graphs of Diameter=2 in Graph Golf (Order,Degree)=(256,23)
17 How to make the (256,23) graph Because We can not use the strong product or copying G 8 method directly in this case (1) The graph obtained from using 32-copies of G 8 is a graph of order = 256 and degree 34 (1) There is no graph pair that their strong product satisfies Order = 256 and Degree 23 We developed another method
18 How to make the (256,23) graph New method : Copying G 8 8 ( i,(t,0)) ( i,(t,1)) G 8 8 = ( i,(m,0)) ( i,(m,1)) ( i,(m,2)) ( i,(b,0)) ( i,(b,2)) ( i,(b,1)) ( i = 1, 2,, 8 )
19 How to make the (256,23) graph New method : Copying G 8 8 ( j, ( i,(t,0))) ( j, ( i,(t,1))) G 8 8 k= ( j, ( i,(m,0))) ( j, ( i,(m,1))) ( j, ( i,(m,2))) ( j, ( i,(b,0))) ( j, ( i,(b,2))) ( j, ( i,(b,1))) ( j = 1, 2,, k i = 1, 2,, 8 )
20 How to make the (256,23) graph New method : Copying G 8 8 Connect k copies of G 8 8 as follows (for all j > l. j, l = 1, 2,, k ) ( j, ( i,(t,0))) ( j, ( i,(t,1))) ( j, ( i,(m,0))) ( j, ( i,(b,1))) ( j, ( i,(m,1))) ( j, ( i,(b,2))) ( j, ( i,(m,2))) ( j, ( i,(b,0))) ( l, ( i,(t,0))) ( l, ( i,(t,1))) ( l, ( i,(m,0))) ( l, ( i,(b,1))) ( l, ( i,(m,1))) ( l, ( i,(b,2))) ( l, ( i,(m,2))) ( l, ( i,(b,0)))
21 How to make the (256,23) graph New method : Copying G 8 8 Connect k copies of G 8 8 as follows (for all j > l. j, l = 1, 2,, k ) ( j, ( i,(t,0))) ( j, ( i,(t,1))) ( j, ( i,(m,0))) ( j, ( i,(b,1))) ( j, ( i,(m,1))) ( j, ( i,(b,2))) ( j, ( i,(m,2))) ( j, ( i,(b,0))) ( l, ( i,(t,0))) ( l, ( i,(t,1))) ( l, ( i,(m,0))) ( l, ( i,(b,1))) ( l, ( i,(m,1))) ( l, ( i,(b,2))) ( l, ( i,(m,2))) ( l, ( i,(b,0)))
22 How to make the (256,23) graph New method : Copying G 8 8 Connect k copies of G 8 8 as follows (for all j > l. j, l = 1, 2,, k ) ( j, ( i,(t,0))) ( j, ( i,(t,1))) ( j, ( i,(m,0))) ( j, ( i,(b,1))) ( j, ( i,(m,1))) ( j, ( i,(b,2))) ( j, ( i,(m,2))) ( j, ( i,(b,0))) ( l, ( i,(t,0))) ( l, ( i,(t,1))) ( l, ( i,(m,0))) ( l, ( i,(b,1))) ( l, ( i,(m,1))) ( l, ( i,(b,2))) ( l, ( i,(m,2))) ( l, ( i,(b,0)))
23 How to make the (256,23) graph New method : Copying G 8 8 Connect k copies of G 8 8 as follows (for all j > l. j, l = 1, 2,, k ) ( j, ( i,(t,0))) ( j, ( i,(t,1))) ( j, ( i,(m,0))) ( j, ( i,(b,1))) ( j, ( i,(m,1))) ( j, ( i,(b,2))) ( j, ( i,(m,2))) ( j, ( i,(b,0))) ( l, ( i,(t,0))) ( l, ( i,(t,1))) ( l, ( i,(m,0))) ( l, ( i,(b,1))) ( l, ( i,(m,1))) ( l, ( i,(b,2))) ( l, ( i,(m,2))) ( l, ( i,(b,0)))
24 How to make the (256,23) graph New method : Copying G 8 8 Connect k copies of G 8 8 as follows (for all j > l. j, l = 1, 2,, k ) ( j, ( i,(t,0))) ( j, ( i,(t,1))) ( j, ( i,(m,0))) ( j, ( i,(b,1))) ( j, ( i,(m,1))) ( j, ( i,(b,2))) ( j, ( i,(m,2))) ( j, ( i,(b,0))) Degree += 4 ( l, ( i,(t,0))) ( l, ( i,(t,1))) ( l, ( i,(m,0))) ( l, ( i,(b,1))) ( l, ( i,(m,1))) ( l, ( i,(b,2))) ( l, ( i,(m,2))) ( l, ( i,(b,0)))
25 How to make the (256,23) graph Copying G 8 8 Copying G 8 Order 64k 8k Degree 4k + 6 k+2 Order/De gree^2 256/Order 64/Order
26 How to make the (256,23) graph Copying G 8 8 Copying G 8 Order 64*4 = 256 8k Degree 4*4 + 6 = 22 k+2 Order/De gree^2 256/Order 64/Order
27 Outline (1) How to make a regular graph (2) Graphs of Diameter=2 in Graph Golf (3) Summary
28 Summary We made new constructing method of graph of diameter = 2. There are many graph constructing method. They give some bounds for Order/Diameter Degree/Diameter Order/Degree problem.
29 Strong product of graphs The strong product G H of graphs G and H is a graph such that the vertex set of G H is the Cartesian product V(G) V(H); and any two distinct vertices (u,u') and (v,v') are adjacent in G H if and only if: u is adjacent to v and u'=v', or u=v and u' is adjacent to v', or u is adjacent to v and u' is adjacent to v Strong product conserves diameter Order(G H )=Order(G)*Order(H) Degree(G H )=Degree(G)Degree(H)+Degree(G)+Degree(H)
30 Constructing methods (Diameter=2) (A) (B) Brown s Construction Generalized Brown s Construction Copying pentagon Copying G 8 Strong product Order Ex. 5k 8k n,m nm Degree k+1 k+2 d 1,d 2 d 1 d 2 +d 1 +d 2 Order/De gree^ /k 8/k (p : a prime k : a natural number )
31 How to make the (64,16) graph Method 1 Order=8 Degree=3 Strong Product Order=8 Degree=3 The graph of Order = 8*8 = 64 Degree = 3* = 15
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