正则图和同构

我想问一下是否已经发布了结果：

我们在两个相连的规则图（假设度为，节点数为）的每对节点之间采用所有可能的不同路径，并记下它们的长度。当然，不同路径的数量是指数的。我的问题是，如果我们对长度进行排序并进行比较（由两个图表获得的列表），并且它们完全相同，我们可以说两个图表是同构的吗？ $d$ $n$

当然，即使这是结果，我们也不能用它来表示图同构，因为如上所述，不同路径的数量是指数的

显然，通过不同的路径，我指的是具有至少一个不同节点的路径。

感谢您的帮助。

— N27
source

在2正则图中，很少的不同路径，因为2正则图是周期的不相交的并集。因此，每对顶点之间有2条或0条路径。

— 森·科恩

这个问题虽然很有趣，但对我来说似乎更适合MathOverflow。

— Niel de Beaudrap 2011年

我相信您的问题的答案是“否”，因为等效条件意味着GI的多项式时间解。

对于图的邻接矩阵，请注意，从到长度为的路径为（允许重复顶点和边）。对于两个图和（具有邻接矩阵和）和，如果对和的元素进行排序，则为了使同构为，这是必要条件所有的列表都相同。 $A$ $G$ $i$ $j$ $k$ $(A^k)_{i, j}$ $G_1$ $G_2$ $A_1$ $A_2$ $k \ge 1$ $A_1^k$ $A_2^k$ $G_1$ $G_2$ $k$

我相信您的猜想等于：

如果和的元素的排序列表对于到（在具有不重复顶点的最长路径上为上界）是相同的，则和是同构的。 $A_1^k$ $A_2^d$ $k = 1$ $n - 1$ $G_1$ $G_2$

因此要解决GI，一个只具有执行的乘法矩阵（和一些额外的时间来排序和比较元素）。这将花费少于时间。 $n - 1$ $n \times n$ $n^2$ $n^4$

我承认我的论点可能存在两个缺陷。首先，GI很有可能具有多项式时间算法，而我们刚刚才刚刚发现它（万岁，我们很有名！）。我发现这种可能性很小。其次（而且更有可能），我所提出的实际上并不等同于您的猜想。

最后的想法。您是否尝试过所有大小为8左右的3个正则图？我认为，如果您的猜想是错误的，那么在尺寸非常小的3正则图中应该有一个反例。

— 贝乔
source

我不知道从i到长度为k的j的不同路径的数量为

。如果是这样，并且如果我对您的工作很了解，那么我的最初假设就会得到回答。

(A^{k})_{i, j}

$(A^{k})_{i,j}$

— N27

@ N27：可以使用矩阵乘法和归纳的定义来证明。

— Tomek Tarczynski 2011年

是的，很容易，其实...

— N27

嗯，看来我的直觉再次使我误入歧途。＃P-complete计算一张图中（甚至只是两个节点之间）不同的简单路径的数量。所以我的说法是错误的，因为它说多项式时间算法等同于计算简单路径。我现在也完全不确定您的猜想是否正确。但是，这有点争议，因为您不太可能选择通过GI解决#P完全问题。

— bbejot 2011年

由于您只比较路径的长度（同时，如果我对您的理解很好，则同时忘记了它们对应的节点对），因此我认为非常相似的图应提供一个反例：最后，您只是在计算固定长度的路径数，独立于它们链接的顶点。例如，我认为这些图形是一个反例：http : //www.mathe2.uni-bayreuth.de/markus/REGGRAPHS/GIF/06_3_3-2.gif和http://www.mathe2.uni-bayreuth.de/ markus / REGGRAPHS / GIF / 06_3_3-1.gif

如果我没记错的话（计算路径很繁琐），它们都具有9个长度为1的路径，18个长度为2的路径，48个长度为3的路径，30个长度为4的路径和36个长度为5的路径

— 阿诺
source

我在第一个图形中计算了36个长度为3的路径，在第二个图形中计算了30个长度为3的图形。问题在于第二个图的长度为3的循环，而第一个图则没有。但是，我仍然同意，应该有一个相对较小的图表作为反例。我还没有找到一个。

— 2011年

我同意你的看法，编写一个测试所有小图形的程序可能会给出一个快速的答案。

— Arnaud，

-2

36
011111111111111100000000000000000000
101111110000000011111111000000000000
110111001100000011000000111111000000
111000101011000010110000110000111000
111000101010100000001110001110000110
111000010110010001001001001001111110001
110110001000011000100101000100101101
110001000101101010000111100000010011
101110100000010100000010100011011011
101001010001100110100000000111100101
100111000001110000011000011000110110
100100010110001110101000101000001110
100010010110001100010110010110110000
1000011010100011010100010001010001000111
100000110001110101000101111100001000
100000001101111001111010000011001000
011100010101000001010100010010001111
011001000000011110011010011100001001
010100100101000100011011100101100100
0101000000101101111000100010010100010
010011000011000101100110001001011100
010010110000101010001001010011011100
010010011000100101111111000100100010011
010001110000011000110100101011100010
001100011001001000100011011111101010010
001100000010111011010100100101010100
001011000011001001001001100110101010
001010100100101001100010111000100101
001010001100100110010101001001101010
001001001100010100101101110010010100
000101100110100000110001001110011001
000101011010100000001110110001101001
000100101001001111111001100001010110001
000010100111010010101100010101000011
00001001101101001001001111111000010000
000001111100010011000010000100111110

011111111111111100000000000000000000
101111110000000011111111000000000000
110111001100000011000000111111000000
111000101011000010110000110000111000
111000101010100001001100001100000111
111000010110010000100011100010001110110
110110001000011000011011010000010110101
110001000101101010011001000001001011
101110100000010100000001011001101011
101001010001100110000010011010010101
100111000001110000100100100100011110
100100010110001110010100010100100110
100010010110001101101000101000011001
100001101010001100000111000111011000
100000110001110101011000100111100000
100000001101111001100111011000100000
011100010101000000001101101110111000
011010000000101100110101110110000001
0101010000101001001011010011101000110010
010100110001001001100010010101010011
010010110000101010000110101001101100
010010000011010111001011010100001100
010001100100010100111100011001010100
010001011000010111100100000010101011
001101000010101001101000010011101100
001100001101000101010110100001001101
001010001100100110101010000101110010
00101000001101101101010000100001011010010
0010011001000110110000010001100100110101
001001011000011000011010111100001010
000100101001001110100101101010000110
0001001001101100110010110010001110001001
0001000110101100100011011100010000001
000011100111000000001110110010100011
000011011011000000110001001101100101
000010111100100001010001010010011110

36
011111111111111100000000000000000000
101111110000000011111111000000000000
110111001100000011000000111111000000
111000101011000010110000110000111000
111000101010100001001100001100000111
111000010010011000100011101011000100
11011000100001010001101000010000010110110
110001000001101110001011110000000011
101110100000011000000001010101101011
101000000001111111010100000111010000
100111000001110000100100101000011101
100100010110001110010100011000100101
100010010110001101101000100100011010
100001101110000100000111000011011001
100001011101100000110001000101100110
100000110101110001001010011010101000
01110001010100000000110110110110001
01101000010010010011010101011010001010
010101000010101001010011001100111000
0101001001010010011001100110010001010110
010010110000100110000110100101101100
010010000111010011011001000001001101
010001110000010100111000011101010001
010001011000011011111100000000000010101011
001101010010100010001000010011011110
001100011001000101010010101001001011
001011000011000101100010010110100101
001010001100101010101010001001111110001
001001100100010111111000001101000110110
001001001100011000011110110100001100
000100101001001111110101001001110001100
0001001001101100110010110010100110000011
000100001010110110101101101110001100000
000011100011001000011100101011100010
000010111000101001010001110010010101
000010011011010010000111011100010010

011111111111111100000000000000000000
101111110000000011111111000000000000
110111001100000011000000111111000000
111000101011000010110000110000111000
111000101010100001001100001100000111
111000010110010000100011001011100100
110110001000011000011010000010011110
110001000001101110010011001100100010
101110100000010100000001100101110011
101001000001101101101000100011010100
100111000001110000100100011000101101
100100010110001110010100101000010101
10001001011000110110100001000010100101010
100001101010001100000111010011001001
100000110101110001010010010110010001
100000011101110010001101100001001010
011100010001000100001101011110011000
0110100001001010001000101101110010000011
010101000110100001011001000001111001
0101001100010010011001100010101001001011
010010100100100110100100110000101011100
010010000011010111001011110000000101
010001110000011000011100110101100100
010001011000010111100100000010110011
001100001101000101010110010001100110
001100000010111011011000110100110101000
001011010011000010010000000111001111
00101001100010101000100010010011001110001
001001100100011011000001011000011110
001001001100010100111010101100001001
000101011010100000100011110100010110
000100101101001010101001000110100101
000100100010110110111000011011000010
000011100111000000001110101010110010
000010111000100101010001101010100101100
000010001011011001110101001101010000

36
011111111111111100000000000000000000
101111110000000011111111000000000000
110111001100000011000000111111000000
111000101011000010110000110000111000
111000101010100001001100001100000111
111000010010011000100011101011000100
11011000100001010001101000010000010110110
110001000001101110011001100001000011
101110100000011000000001010101101011
101000000001111111000110010110010000
100111000001110000100100101000011101
100100010110001110010100011000100101
100010010110001101101000100100011010
100001101110000100000111000011011001
100001011101100000100011010100100110
100000110101110001011000001011101000
01110001010100000000110110110110001
01101000010010010011010101011010001010
010101000010101001001011011000111000
010100110001000101101000110110001001110
010010110000100110100010001101110001
010010000111010011010011000100001101
010001100100011000111100010101010100
010001011000011011111100000000000010101011
0011010100101000100100000001000111011110
001100001101001001110010001001010011
001011000011000101101000010011100101
001010001100101010001110100001101100
001001100100010111111000001101000110110
00100101100001010001101011110000100001
000100101001001111110101001001110001100
0001001001101100110010101010110010000011
000100001010110101110101100101100000
000011100011001000010110101110100010
000010111000101001010001110010010101
0000100110110100100011010110010010010

011111111111111100000000000000000000
101111110000000011111111000000000000
110111001100000011000000111111000000
111000101011000010110000110000111000
111000101010100001001100001100000111
111000010010011000100011101011000100
11011000100001010001101000010000010110110
110001000001101101011001100001100001
101110100000011000000001010101101011
101000000001111111000110010110010000
100111000001110000100100101000011101
100100010110001110010100011000100101
100010010110001101101000100100011010
100001101110000100000111000011011001
100001011101100000100011010100100110
100000110101110010011000001011001010
011100000101000100101101001110101000
011010010100100000010101110010010011
010101000010101010001011011000011010
010100110001000101101000110110001001110
010010110000100110100010001101110001
010010000111010011010011000100001101
010001100100011000111100010101010100
010001011000011011111100000000000010101011
001101010010100001010000000111111100
001100001101001001110010001001010011
00101100001100011010100001000011000111
001010001100101010001110100001101100
001001100100010111111000001101000110110
00100101100001010001101011110000100001
0001001110010010100010001001100110010101
000100100110110001101010110010100001
000100001010110110110101101100101000010
000011100011001000010110101110100010
0000101010001011011100110011011010001100
000010011011010001001101011001110000

— trg787
source

在所有这些图中，lambda = mu

— trg787 2011年

这是3个最简单的对（非同构）

— trg787

那是什么？！！您怎么知道至少有一条不同的道路？

— N27

我的意思是你怎么知道每对节点之间所有可能路径的列表是相同的？

— N27

无论如何，对不起，我不明白您已经测试过或试图说什么...我的问题是，对于2个非同构图，所有节点对之间的不同路径的所有长度的2个列表是否不同？

— N27