CC BY
9
2021 Прикладная теория графов № 54
УДК 519.176 DOI 10.17223/20710410/54/6
SERIES OF FAMILIES OF DEGREE SIX CIRCULANT GRAPHS1
E. A. Monakhova
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk,
Russia
E-mail: [email protected]
An approach for constructing and optimizing graphs of series of analytically described circulant graphs of degree six with general topological properties is proposed. The paper presents three series of families of undirected circulants having the form C(N(d, p); 1, s2(d,p), s3 (d,p)), with an arbitrary diameter d > 1 and a variable parameter p(d), 1 ^ p(d) ^ d. The orders N of each graph in the families are determined by a cubic polynomial function of the diameter, and generators s2 and s3 are defined by polynomials of the diameter of various orders. We have proved that the found series of families include degree six extremal circulant graphs with the largest known orders for all diameters. By specifying the functions p(d), new infinite families of circulant graphs including solutions close to extremal graphs are obtained.
Keywords: Abelian Cayley graph, degree/diameter problem, families of degree six circulant graphs, triple loop graphs, extremal circulant graphs.
СЕРИИ СЕМЕЙСТВ ЦИРКУЛЯНТНЫХ ГРАФОВ СТЕПЕНИ ШЕСТЬ
Э. А. Монахова
Институт вычислительной математики и математической геофизики СО РАН,
г. Новосибирск, Россия
Предложен подход к построению и оптимизации графов серий аналитически описываемых циркулянтных графов степени шесть с общими топологическими свойствами. Представлены три серии семейств неориентированных циркулянтов вида C(N(d,p); 1, s2(d,p), s3(d,p)) произвольного диаметра d > 1 с переменным параметром p(d), 1 ^ p(d) ^ d. Порядки N каждого графа в семействах определяются кубическим полиномом от диаметра, а образующие s2 — полиномами от диаметра различных порядков. Доказано, что найденные серии семейств включают экстремальные циркулянтные графы степени 6 с самыми большими известными порядками для всех диаметров. Посредством задания функций p(d) построены новые бесконечные семейства циркулянтных графов, включая решения, близкие к экстремальным графам.
Ключевые слова: граф Кэли абелевой группы, проблема d/k графов, семейства циркулянтных графов степени 6, трёхмерные кольцевые циркулянтные графы, экстремальные циркулянтные графы.
1The work was supported by ICMMG SB RAS budget project no. 0251-2021-0005.
1. Introduction
Circulant graphs are Cayley graphs of Abelian groups. They are widely studied in
graph theory and discrete mathematics and play an important role in various applications
including network design, see surveys [1-4] and references in them.
Let si,s2,...,sk, N be a set of integers such that 1 ^ s1 < s2 < ... < sk < N,
and let S = (s1, s2,..., sk) be a generator set. An undirected graph C(N; S) with sets of
vertices V = {0,1,..., N — 1} and edges E = {(v, (v ± si) mod N) : v E V, l = 1,..., k},
is called circulant graph, k is the dimension, N is the order of the graph. The diameter
of C is dia(C) = maxD(i, j), where D(i, j) is the length of a shortest path from vertex i i,jev
to vertex j. In the paper, we consider the class of even-degree multi-loop circulants when s1 = 1 and sk = N/2 for even N [1-7].
For any given k and d, the function M(d, k) defines the maximum number of vertices that can be reached from any vertex of a circulant graph of dimension k in at most d steps. It is known [8] that
M(d, k) = 2i k d
i=0
where the value of M(d, k) can be viewed as the Moore bound for circulant graphs of dimension k. This is a polynomial in d of order k:
2k 2k—1
M(d, k) = — dk + --— dk—1 + O(dk—2).
k! (k — 1)!
Following [5], we define for any natural d and k the extremal function P(d, k) as maximum possible (attainable) natural number N such that there is a set of generators S = (1,s2,...,sk) for which dia(C (N; S)) ^ d. We have P (d, k) = M (d, k) for k ^ 2, namely,
P(d, 1) = 2d +1, P(d, 2) = 2d2 + 2d + 1, and P(d,k) < M(d,k) for k > 2 [5, 9-11], including
4d3 od
P(d, 3) < M(d, 3) = -d- + 2d2 + Od + 1.
3 3
It is difficult to obtain the exact value of P(d, k) for k ^ 3 and large d. This is a solution to the degree/diameter problem [12-14] in the class of circulant networks and it leads to an exhaustive computer search. The lower bounds for P(d, k) in each individual case may depend on the diameter under consideration and are usually obtained by finding infinite families of graphs with these estimates (see also the table of the orders of the largest known circulant graphs [15]). Note that for a given dimension and diameter, circulant graphs with the maximum possible order (or the nearest to it), taken as a model of interconnection networks of multiprocessor systems, have the maximum reliability and connectivity and the minimum number of steps in the implementation of routing algorithms. Using circulant graphs as a network-on-chip (NoC) topology [16, 17] becomes relevant due to their better structural properties and high scalability compared to standard NoC topologies (2D-mesh, 2D-torus). Three-dimensional circulant topologies for NoCs [18, 19] are a promising alternative to the classic 3D-mesh and 3D-torus topologies [20, 21] due to the best characteristics of the diameter and average distance between nodes. An urgent task for NoCs with three-dimensional circulant topology is the development of effective routing
algorithms related to the peculiarities of the requirements for the use of chip resources in NoCs.
The following infinite families of degree 6 circulant graphs with an analytical description are known in the literature. The family of circulant graphs of diameter d
C(3d2 + 3d + 1; 1, 3d + 1, 3d + 2), d ^ 1,
was obtained in [22] as a solution to the optimization problem for hexagonal tessellation on the plane of circulants of the form C(N; si, s2, s1 + s2) with diameter d. Note that the family is a subset of the family of Eisenstein — Jacobi graphs [23] introduced for constructing perfect codes. Shortest path search algorithms for this family were given in [24, 25]. In [26], the family of three-dimensional circulants has been found as a solution to the optimization problem when considering the Kronecker product of two circulants of degrees 2 and 3. The family has diameter d = 3,5 (mod 6) and order N = 4d2 — 2d — 2.
Families of the so-called multiplicative circulant graphs of dimension k ^ 3 have been obtained in [5, 6, 8, 27]. In [6, 27], the diameter and shortest path search algorithms were also given for the families. Below for k = 3, we give the orders of graphs of these families represented as a function of diameter d; for diameter d ^ 3, d = 0 (mod 3) [8, 27]:
2d + 3 \3 8 ,3 2
N = 37 d3 + O(d2);
for diameter d ^ 5, d = 2 (mod 3) [6]:
n = (^) =27 d3 + o<d2);
and for diameter d ^ 3, d = 0 (mod 3) [5]:
32 8 2 N = — d3 + - d2 + - d. 27 9 3
The families of the largest known degree 6 circulant graphs (extremal circulants) were discovered by E. A. Monakhova [9] and in [28] (for Cayley graphs of Abelian groups).
For dimension k > 3, R. R. Lewis have recently obtained families of the largest known circulant graphs of degrees 8 and 10 with an analytical description [10, 11]. Formulas for the order N in diameter d ^ 3 of degree 8 graphs are shown below:
N_ 'd4/2 + d3 + 3d2 + 2d for d = 0 (mod 2),
= d4/2 + d3 + 3d2 + 3d +1/2 for d = 1 (mod 2).
As we can see, only individual families of circulants have been obtained earlier, and their topological properties and the possibility of obtaining good routing algorithms for some of them have been studied.
In [29], the author introduced the concept of a series of families of three-dimensional circulant graphs defined by two parameters, one of which is their common diameter and the second is a function of the diameter. This paper is an extension of the previous one. We introduce three series of parametrically described infinite families of circulant graphs that differ in the set of graph generators. We prove that these series include extremal triple loop graphs with the maximum number of vertices for any diameter and have common
topological and communicative properties (Sections 2-4). This result makes it possible to create previously unknown infinite families of dense circulant graphs with varying the diameter and also to construct a series of d graphs for a fixed diameter d > 1. Section 5 presents examples of constructing new infinite families of triple loop networks in all the series based on specifying functions p = p(d). The results of constructing the three series of circulants are presented in Section 6 and the Appendix. Another important property is the existence of a general structure of the graphs of the found families, which leads to the construction of a general analytical method for finding the shortest paths and routing algorithm for them.
2. A series of circulants with generators s2 = f (d)
Consider a set of circulant graphs of the form C(N; 1, s2, s3) with s1 = 1 and 1 < s2 < < s3 — |_N/2_|. Denote A = s3 — s2. We place the vertices of a graph C on the line, as shown in Fig. 1 and 2. They are labeled from 0 to N (the vertex N is understood to be the same vertex as 0). Define +s3- and —s3-jumps from i, if we travel to (i + s3) mod N or (i — s3) mod N, respectively. Similarly, define +s2- and —s2-jumps and +1- and — 1-jumps. In the paper, we also use ±A-jumps of length 2. Since the circulant graphs are vertex transitive, it is sufficient to consider 0 as the initial vertex. For simplicity, we will use the notation D(v) instead of D(0,v).
The following slightly modified Lemma from [29] is used in the proofs of Theorems.
Lemma 1 [29]. In a triple loop graph C(N; 1,s2,s3), let 0 — i < j < N be any two vertices, and let D(i), D(j) be the values of their distances from 0. Then, using +1-jumps from i and —1-jumps from j, we have
max D(v) = L(j — i + D(i) + D(j))/2j.
i—v—j
The maximum is reached at the vertex v = [(j + i + D(j) — D(i))/2j.
At first, we will consider the case when generators s2 and s3 of C(N;1,s2,s3) are polynomials in d of orders 1 and 2, respectively. We have the following result.
Theorem 1. Let 1 — p — d for any integer d > 1. Then any circulant graph C(N; 1, s2, s3), where
N = p3 — (4d +1 /2)p2 + 4d2p + 2d2 + 2d +1 N = p3 — (4d +1 /2)p2 + 4d2p + 2d2 + 2d +1 /2 s2 = 2(d — [p/2]) + 1,
s3 = 2(d — [p/2j)s2 + 1,
has diameter d.
Proof. Consider a graph C(N; 1,s2,s3), where N, s2 and s3 are given by (1), d > 1 and 1 — p — d. The representation of vertices of C is shown in Fig. 1.
From (1) it follows that r = N mod s3 = (s3 + s2)/2 and hence
N = ps3 + (s3 + s2)/2. (2)
We need to prove that the graphs given by (1) have the diameter d. We will divide all the vertices 0 — v — N into the following segments Ai = [is3, (i + 1)s3], i = 0,1,... ,p, which in turn are divided into the subsegments:
for even p, for odd p,
Ai = [is3, N — (p — i)s3], A = [N — (p — i)s3, (i + 1)s3].
Fig. 1. The representation of vertices of a graph defined by (1)
We have
D(iss) = i, D((i + 1)sa) = i + 1, D(N - (p - ¿)ss) = p - i.
All vertices in A\ can be reached from is3 by +s2-jumps and ±1-jumps or from N — (p — i)s3 by —s2-jumps and ±1-jumps. Similarly, all vertices in A[ can be reached from N — (p—i)s3 by +s2-jumps and ±1-jumps or from (i + 1)s3 by —s2-jumps and ±1-jumps. From the structure of the graphs, it follows that |Ai| = |Ar| + s2, and in Ai distances D(v) of vertices v are the same as in [is3 + s2, N — (p — i)s3]. Hence, it is sufficient only to consider the segments Ai. In turn, they are divided into the subsegments
Aik = [is3 + ks2, iS3 + ks2 + [^2/2]],
Arik = [is3 + ks2 + [S2/2], is3 + (k + 1)s2],
where k = 0,1,...,d — |_p/2j, since we have |_(s3 + s2)/2s2j = d — |_p/2j by (1). Note that for k = d — |_p/2j the segment Aik is not considered. We have |A|fc| = [s2/2] = = |Aik | + 1 and accordingly in Aik the values of D(v) of the vertices v are the same as in [is3 + ks2 + 1, is3 + ks2 + [s2/2]]. Hence, it is sufficient only to consider the segments A'fc to determine the maximum distance in the graph. We have
D(is3+ks2) = i+k, D(is3+ks2 +[^2/2]) = d+[p/2]—i — k, D(is3 + (k+1)s2) = i+k+1. To obtain in Aik the maximum distance of vertices from 0, we use Lemma 1: max D(v) = |_(i + k + d + [p/2] — (i + k) + 1^/2] )/2j = d.
Now we will show that there is a vertex v at the distance D(v) = d from 0. Let v = N —
— ps3 — (d — p)s2. For it, as we can see, there is a path with the number of steps d using
— s3-jumps and —s2-jumps. On the other hand, using (2) and substituting s3 from (1) into v, we obtain v = |_s3/2j + [s2/2] — (d — p)s2 = [p/2]s2 + d — [p/2] + 1. It follows that the length of the path from 0 to v through the nearest vertex of the form [p/2]s2 is d + 1. Similarly, we have v = ([p/2] + 1)s2 — (d — [p/2]) and hence the length of the path from 0 to v through another nearest vertex ([p/2] + 1)s2 is also equal to d +1. The lengths of other paths from 0 to v are greater than d. Therefore, the graphs prescribed by (1) have diameter d. ■
We introduce the concept of a series of graphs of r. For any integer d > 1, let p = = 1, 2,..., d. Then we have an infinitive set (series) r of triple loop graphs with diameters d =2, 3,...:
r= u C (N ;1,S2,S3), where N, s2 and s3 are defined using (1).
Now consider the optimization problem for r: to find a function p = p(d) giving the maximum of N = N(p) for all d > 1 among graphs of r with diameter d > 1.
Theorem 2. Let d > 1 be an integer. The maximum order of circulant graphs C(N; 1, s2, s3) G r with diameter d is
{32d3/27 + 16d2/9 + 2d +1 for d = 0 (mod 3),
32[d/3j3 + 48[d/3j2 + 26[d/3j +5 for d = 1 (mod 3), (3)
32[d/3j3 + 80[d/3j2 + 70[d/3j +21 for d = 2 (mod 3).
The maximum is achieved with the following generators:
{(4d/3 + 1,16d2/9 + 4d/3 + 1) for d = 0 (mod 3),
(4[d/3j+2±1,16[d/3j2+16[d/3j+5±(4[d/3j+2)) for d = 1 (mod 3), (4) (4[d/3j +3,16[d/3j2 + 28[d/3j +13) for d = 2 (mod 3).
Proof.
Case 1. Let p be an even number. Our goal is to find the maximum function N = N(p) for a given d > 1. The function N is a cubic polynomial in p with the following coefficients: a = 1 > 0, b = —4d — 1/2, c = 4d2. We will find a valued-integer function p(d) such that N defined by (1) has a maximum for any d > 1. The discriminant D = 3ac — b2 = = — 4d2 — 4d — 1/4 < 0. Hence, N(p) has one maximum when
p = —b + ^ = 4 d — 2. /i+d^L + 1.
1 3a 3 3V 16 6
_ 2 1 2
Since d +1/4 < -i/d2 + d +1/16 < d + 1/2, we have -d--< p < -d. Since p(d) is a
3 6 3
valued-integer function with even values, we take the nearest even integer and obtain
p =2[d/3j, if d = 0,1 (mod 3).
Substituting the found p into (1) we obtain s2, s3 and N for d = 0,1 (mod 3) (the version with " +").
Case 2. Let p be an odd number. Similarly to case 1, we have the following: the maximum of N = N(p) for a given d is reached when
p = 2[d/3j +1, if d = 1, 2 (mod 3).
Substituting the found p into (1), we obtain s2, s3, and N for d = 1, 2 (mod 3) (the version with "—"). ■
3. A series of circulants with generators s2 = f (d2)
We will now study the case when generators s2 and s3 of a graph are polynomials in d of order 2. We have the following result.
Theorem 3. Let 1 — p < d for any integer d > 1. Then any circulant graph C(N; 1, s2, s3), where
'N = — 8p3 + (8d + 4)p2 + 2d + 1,
s2 = — 4p2 + 4dp + 4p — 2d, (5)
_s3 = s2 + 4d — 4p + 2,
has diameter d.
We introduce the concept of a series of graphs of $. For any integer d > 1, let p = 1, 2,... ,d — 1. Then we have an infinite set (series) $ of triple loop graphs with diameters d =2, 3,...:
$ = U C (N ;1,s2,s3),
(d>1)A(1— p<d)
where N, s2 and s3 are defined by (5).
First, we prove Lemma 2, which gives a general analytical method for obtaining the distance function D(v) for all graphs in $. We can restrict our consideration to the values 0 — v — |_N/2j, since D(v) = D(N — v) in any circulant graph.
Lemma 2. Let C(N; 1, s2, s3) G $ be a triple loop graph and 5 = 2(d +1 — p). For any vertex v, 0 — v — |_N/2j of the graph C let i = |_v/s3j. If v — (i + 1)s2, then
D(v) = <
i + 2 j + 5 — |k|, if (0 — i + 2 j — d — 5) and (—5 — k < 5 — 2),
or if (d — 5 < i + 2j — d) and (—5 — k — d — 5 — (i + 2j)), or (6)
i + 2j — (d — 5) — k < 5 — 2, 2d +1 — (i + 2 j + 5 — |k|) otherwise,
where
j = L(v — is3)/(s3 — s2)j, k = v — is3 — j(s3 — s2) — 5. If v > (i + 1)s2, then
D(v)
'i + 5 - |k|, if (0 ^ i ^ d - 5) and (-5 < k < 5 - 1),
or if (d - 5<i ^ d - 5/2) and (-5 < k ^ d - 5 - i), or i - (d - 5) ^ k < 5 - 1, ^2d +1 - (i + 5 - |k|) otherwise,
where
j = L(v — (i + 1)s2)/(s3 — s2)j, k = v — (i + 1)s2 — j (s3 — s2) — 5 +1. (9)
Proof. Consider a graph C(N; 1, s2, s3) G $, where d > 1 and p = 1, 2,..., d — 1. For simplicity, we define A = s3 — s2 = 4(d — p) + 2, 5 = A/2 + 1. Hence, p = |_s3/Aj, N = (2p — 1)s3 + A + 1, and
5 = 2(d +1 — p). (10)
The representation of vertices of C is shown in Fig. 2, here q = 2p — 1, r = A + 1. We will divide the vertices 0 — v — N/2 into the following segments:
A = [is3, (i + 1)s2], Bi = [(i + 1)s2, (i + 1)s3], 0 — i<p.
Note that v = |_N/2j G Bp-i. We have |Ai| = (p — i — 1)A + A/2 + 1, |Bi| = (i + 1)A, D(is2) = D(is3) = i.
Consider two types of vertices in Ai reached from vertices is3 by +A-jumps and from vertices (i + 1)s2 by —A-jumps:
Xij = is3 + j A = (i + j )s3 — js2, xj = Xj + 5 = (i + 1)s2 — (p — 1 — i — j )A = (p — j )s2 — (p — 1 — i — j)s3,
Fig. 2. The representation of vertices of a graph defined by (5)
where 0 ^ i < p, 0 ^ j < p — i. Using (10) and taking into account that there is a path from Xj to Xj with length 2p — 1 — (i + 2j) + 8 = 2d +1 — (i + 2j), we obtain D(xj) = min{i + 2j, 2d +1 — (i + 2j)}. Therefore,
i + 2j, if i + 2j ^ d, 2d +1 — (i + 2j), if i + 2j > d.
D(Xij)= T^' "'T ^ (11)
Similarly, we have D(xj) = min{2d +1 — i — 2j — 8, i + 2 j + 8} and
D ij)= f + 2j + 8 if i + 2j i d - 8, (12)
v j [2d +1 — (i + 2j + 8), if i + 2 j > d — 8. v 7
Consider the vertices in Bi reached from vertices (i + 1)s2 by +A-jumps:
yij = (i + 1)S2 + j A = (i + 1 — j)s2 + jsa, 0 ^ i<p, 0 ^ j ^ i +1. From here we have
D(y ij) = i + 1. (13)
i
Note that Bi = (J [y^ ,yj+i]. j=0
Let 0 ^ v ^ |_N/2J be a vertex of the graph. Compute i = |_v/s3J. Case 1. Let v ^ (i + 1)s2, hence v G Ai, 0 ^ i < p. Compute j and k by (7). Having 0 ^ j < p — i, —8 ^ k < 8 — 2, and using (10), we obtain 0 ^ i + 2j ^ 2d — 8 — i. Taking into account that either 0 ^ i + 2j ^ d — 8, or d — 8 < i + 2j ^ d, or d < i + 2j ^ 2d — 8 — i, and using (11) and (12), we obtain (6).
Case 2. Let v > (i + 1)s2, hence v G Bi, 0 ^ i < p. Compute j and k by (9). 2a) Let 0 ^ i ^ d — 8. Using (13), we obtain D(v) = i + 8 — |k| when —8 < k < 8 — 1. 2b) Let i > d — 8. In the graph, the movement from N to 0 alone generators —s3 and — s2 decreases the function D(v) for vertices v G [yij ,yij+1], d — 8<i<p, 0 ^ j ^ i. Consider the vertices mij lying in the middle of segments [yij ,yij+1]:
mij = yij + 8 — 1 = (i + 1)s2 + j A + 8 — 1, d — 8<i<p, 0 ^ j ^ i.
The movement from N to 0 alone generators — s3 and — s2 gives
D(mij ) = 2d +1 — i — 8, (14)
since mij = N — (d + 1 — 8/2 — j)s3 — (d — 8/2 — i + j)s2. Taking into account that either —8 <k ^ d — i — 8, or |k| < i + 8 — d, or i + 8 — d ^ k<8 — 1, and using (13) and (14) and taking the minimum path from 0 to v, we obtain (8). ■
We can now prove Theorem 3.
Proof. Using the formulas (6) and (8), we have the following: the sum of lengths of two possible paths from 0 to v is always equal to 2d +1 for any vertex v. Therefore, one of the lengths is less than or equal to d. Now it is necessary to show that there is a vertex v of C with D(v) = d. Let v = |_N/2j = —4p3 + (4d + 2)p2 + d. For even p, there is a path of length d from 0 to v, since |_N/2j = d — p + (p/2)(s2 + s3). For odd p we have |_N/2j — N = —[N/2] = d — p — |_p/2j s2 — [p/2] s3 and for v there is also a path of length d from 0. The same result follows from (8). There are no other paths in the graph to v with length less than d. ■
Now, let us consider the optimization problem for $: to find a function p = p(d) giving the maximum of N = N(p) for all d > 1 among graphs of $ with diameter d > 1.
Theorem 4. Let d > 1 be an integer. The maximum order of circulant graphs C(N; 1, s2, s3) G $ with diameter d is
| 32d3/27+ 16d2/9 +2d + 1 for d = 0 (mod 3),
N = ! 32|_d/3_|3 + 48|_d/3_|2 + 30Ld/3j + 7 for d = 1 (mod 3), (15
1 32 |_d/3j3 + 80 Ld/3J2 + 70 Ld/3j + 21 for d = 2 (mod 3).
The bound is achieved with the following generators:
{(8d2/9 + 2d/3, 8d2/9 + 2d + 2) for d = 0 (mod 3),
(8|_d/3j2 + 6Ld/3j +2, 8|_d/3j2 + 10 Ld/3j +4) for d = 1 (mod 3), (16) (8|_d/3j2 + 10Ld/3j +4, 8|_d/3j2 + 14[d/3j +6) for d = 2 (mod 3).
Proof. Consider a circulant graph C(N; 1, s2, s3) G $ with diameter d. The function N is a cubic polynomial in p with the following coefficients: a = —8, b = 8d + 4, c = 0. We will find a valued-integer function p(d) such that N defined by (5) has a maximum for any d > 1.
Take the derivative of N(p) with respect to p: —— = —24p2 + 2(8d + 4)p = 0. Therefore,
dp
N has the maximum when p = (2d + 1)/3. Since p(d) is a valued-integer function, we take the nearest integer and obtain for any d ^ 1
p(d) = p* = (2d + d mod 3)/3.
Substitutingp* into (5), we get that N, s2 and s3 are equal to (15) and (16), respectively. ■
The family with p = d, namely the graphs C(4d2 + 2d + 1; 1, 2d, 2d + 2), where d > 1, can also be included in $ since they have diameter d.
4. A series of circulants with generators s2 = f (d3)
The case of circulant graphs, when all N, s2 and s3 are polynomials in d of order 3, was studied in [29].
Theorem 5 [29]. Let 1 — p < d for any integer d > 1. Then any circulant graph C(N; 1, s2, s3), where
N = 8p3 — (16d + 8)p2 + (8d2 + 8d)p + 2d +1, s2 = 4p(d — p)2 + 2p(d — p) + d — 3p, (17)
s3 = s2 + 4p,
has diameter d.
For any integer d> 1, let p = 1, 2,...,d — 1. of triple loop graphs with diameters d = 2, 3,...:
Then we obtain an infinite set (series) ^
*= U C (N ;1,S2,S3),
(d>1)A(1<p<d)
where N, s2 and s3 are defined by (17). In [29], the optimization problem for ^ has been also solved.
Theorem 6. Let d > 1 be an integer. The maximum order of circulant graphs C(N; 1, s2, s3) G ^ with diameter d is
(32d3/27 + 16d2/9 + 2d +1 for d = 0 (mod 3),
32|_d/3j3 + 48|_d/3j2 + 22|d/3j +3 for d = 1 (mod 3), 32 Ld/3j3 + 80|d/3j2 + 70|d/3j +21 for d = 2 (mod 3).
18)
The bound is achieved with the following generators:
(16d3/27+4d2/9,S2+4d/3) (16d3/27+4d2/9—2d/3+17/27, S2+4d/3—4/3) (s2, S3) =4 or
(16d3/27+4d2/9—4d/3—46/27, S2+4d/3+8/3) (16d3/27+4d2/9—2d/9—29/27, S2+4d/3+4/3)
for d = 0 (mod 3),
for d = 1 (mod 3), for d = 2 (mod 3).
19)
For all diameters d = 0, 2 (mod 3) the value of N from (18) is equal to the maximum N from (15), but for d = 1 (mod 3) it is less. Note that for d = 1 (mod 3) there are two sets of generators for N, defined by (18). It is easy to check that for d = 0, 2 (mod 3) the generators (19) are obtained by an equivalent transformation of generators (4), and thus the graphs from Theorems 2 and 6 are isomorphic for these diameters.
Comparing the values of N obtained in Theorems 2, 4, and 6, we come to the conclusion that (15) gives the largest values of N among graphs of sets r, $ and ^ for all diameters d. This result is in good agreement with the extremal values of N obtained in [9, 28].
5. New families of three-dimensional circulants
If we take any valued-integer functions p(d) such that 1 ^ p(d) ^ d as a parameter p, then we can generate new infinite families of circulant graphs. We have shown examples of new families of circulants constructed by this method for sets r, $ and ^ (Theorems 2, 4, and 6). These found families are the best ones in the ratio N/d (order/diameter) among all known families of degree 6 circulants. Some examples of constructing other possible families of sets r, $, and ^ are presented below.
Example 1. Let C(N; 1, s2, s3) G r, d > 1, and
p(d)
J2|_d/3j — 1 for d = 0 (mod 3), |2(Ld/3j +1) for d = 1, 2 (mod 3).
Substituting the value of p in The resulting N is less than (3) for d = 1 (mod 3).
1), we obtain N, s2 and s3 for a new infinite family. by 4|d/3j +2 for d = 0, 2 (mod 3) or by 12|d/3j + 2
Example 2. Let C(N; 1, s2, s3) G $, d > 1, and
2d/3 + 1 for d = 0 (mod 3), p(d)=<(2(d — 1)/3 for d = 1 (mod 3), (2d — 1)/3 for d = 2 (mod 3).
Substituting the value of p in (5), we obtain N, s2, and s3 for a new infinite family. Resulting N is less than (15) by 8[d/3j +4 for d = 0, 2 (mod 3) or by 24[d/3j + 4 for d = 1 (mod 3).
With respect to N, the resulting graphs C(N;1,s2,s3) in examples 1 and 2 are the nearest to the graphs with the maximum N among all graphs of the sets r and $, respectively.
For the following family was created in [29].
Example 3. Let C(N; 1, s2, s3) G d > 1, and p(d) = [d/2]. Then
. i(d3 + 2d2 + 2d + 1; 1, (d3 + d2 — d)/2, (d3 + d2 + 3d)/2) for even d,
( ;,s2,s3) [(d3 + d2 + d;1, (d3 — 3)/2 — d, (d3 — 3)/2 + d + 2) for odd d.
For the following function p(d) generates a new family of circulants.
Example 4. Let C(N; 1, s2, s3) G d > 1, and
J 2 |_d/3j for d = 0 (mod 3), = |2|_d/3j +1 for d = 1, 2 (mod 3).
Substituting the value of p in (17), we obtain N, s2, and s3 for a new infinite family.
As for the ratio N/d, the new families of circulants in examples 1 and 2 are better than families in [5, 6, 8, 22, 26], and the family in example 3 is better than families in [6, 8, 22, 26] and, moreover, in contrast to families from [5, 6, 8, 26], all the obtained families exist for any diameter of the graph.
6. Descriptions of graphs series
Using the system Wolfram Mathematica 10, fragments of circulant families for all sets r, $, and ^ with diameters d = 2,..., 25 have been implemented (see the Appendix).
Tables 1-3 show part of the descriptions of the obtained circulants C(N; 1, s2, s3) with diameters d, 2 — d — 10: the diameters of graphs d, the values of p, where 1 — p — d for r, $ and 1 — p — d — 1 for the orders of graphs N, and generators s2 and s3.
Figures 3-5 show the dependence of the number of vertices N on diameter d, 2 — d — 25, and parameter p for the fragments of descriptions for sets r, $, and The values of N of graphs corresponding to the families with the maximum orders for the given diameters are marked in blue in Fig. 3-5. Figure 5 additionally shows the orders of graphs of the family from Example 4 in red. Note that parameter p has a different meaning in figures: p = |_N/s3j in Fig. 3, p = Ls3/(s3 — s2)j in Fig. 4, and p = (s3 — s2)/4 in Fig. 5.
Table 1
Descriptions of the graphs of r
d P N S2 S3 d P N S2 S3
2 1 21 3 13 8 1 369 15 241
2 2 19 3 7 8 2 535 15 211
3 1 49 5 31 8 3 647 13 183
3 2 55 5 21 8 4 713 13 157
3 3 47 3 13 8 5 737 11 133
4 1 89 7 57 8 6 727 11 111
4 2 111 7 43 8 7 687 9 91
4 3 111 5 31 8 8 625 9 73
4 4 97 5 21 9 1 469 17 307
5 1 141 9 91 9 2 691 17 273
5 2 187 9 73 9 3 851 15 241
5 3 203 7 57 9 4 957 15 211
5 4 197 7 43 9 5 1013 13 183
5 5 173 5 31 9 6 1027 13 157
6 1 205 11 133 9 7 1003 11 133
6 2 283 11 111 9 8 949 11 111
6 3 323 9 91 9 9 869 9 91
6 4 333 9 73 10 1 581 19 381
6 5 317 7 57 10 2 867 19 343
6 6 283 7 43 10 3 1083 17 307
7 1 281 13 183 10 4 1237 17 273
7 2 399 13 157 10 5 1333 15 241
7 3 471 11 133 10 6 1379 15 211
7 4 505 11 111 10 7 1379 13 183
7 5 505 9 91 10 8 1341 13 157
7 6 479 9 73 10 9 1269 11 133
7 7 431 7 57 10 10 1171 11 111
P
Fig. 3. Number of vertices N = N(d,p) for C(N; 1, s2, s3) G r
Table 2
Descriptions of the graphs of $
d P N S2 S3 d P N S2 S3
2 1 17 4 10 8 1 77 16 46
2 2 21 4 6 8 2 225 40 66
3 1 27 6 16 8 3 413 56 78
3 2 55 10 16 8 4 593 64 82
3 3 43 6 8 8 5 717 64 78
4 1 37 8 22 8 6 737 56 66
4 2 89 16 26 8 7 605 40 46
4 3 117 16 22 8 8 273 16 18
4 4 73 8 10 9 1 87 18 52
5 1 47 10 28 9 2 259 46 76
5 2 123 22 36 9 3 487 66 92
5 3 191 26 36 9 4 723 78 100
5 4 203 22 28 9 5 919 82 100
5 5 111 10 12 9 6 1027 78 92
6 1 57 12 34 9 7 999 66 76
6 2 157 28 46 9 8 787 46 52
6 3 265 36 50 9 9 343 18 20
6 4 333 36 46 10 1 97 20 58
6 5 313 28 34 10 2 293 52 86
6 6 157 12 14 10 3 561 76 106
7 1 67 14 40 10 4 853 92 118
7 2 191 34 56 10 5 1121 100 122
7 3 339 46 64 10 6 1317 100 118
7 4 463 50 64 10 7 1393 92 106
7 5 515 46 56 10 8 1301 76 86
7 6 447 34 40 10 9 993 52 58
7 7 211 14 16 10 10 421 20 22
Fig. 4. Number of vertices N = N(d, p) for C(N; 1, s2, s3) E $
Table 3
Descriptions of the graphs of ^
d P N S2 S3 d P N S2 S3
3 1 55 20 24 8 3 737 329 341
3 2 39 9 17 8 4 657 284 300
4 1 105 43 47 8 5 497 203 223
4 2 105 38 46 8 6 305 110 134
4 3 57 13 25 8 7 129 29 57
5 1 171 74 78 9 1 595 278 282
5 2 203 83 91 9 2 915 423 431
5 3 155 56 68 9 3 1027 468 480
5 4 75 17 33 9 4 979 437 453
6 1 253 113 117 9 5 819 354 374
6 2 333 144 152 9 6 595 243 267
6 3 301 123 135 9 7 355 128 156
6 4 205 74 90 9 8 147 33 65
6 5 93 21 41 10 1 741 349 353
7 1 351 160 164 10 2 1173 548 556
7 2 495 221 229 10 3 1365 631 643
7 3 495 214 226 10 4 1365 622 638
7 4 399 163 179 10 5 1221 545 565
7 5 255 92 112 10 6 981 424 448
7 6 111 25 49 10 7 693 283 311
8 1 465 215 219 10 8 405 146 178
8 2 689 314 322 10 9 165 37 73
P
Fig. 5. Number of vertices N = N(d,p) for C(N; 1, s2, s3) e ^
7. Conclusion
The graphs of three series of the circulants obtained in this paper have not only general topological properties, but also general communicative properties that relate to the organization of routing for these graphs. We do not review in detail the existing routing algorithms for degree 6 circulant graphs. This is a topic for a separate paper. Let's just say that few routing algorithms are known for these graphs and they are not efficient for use in NoCs, therefore analytical routing algorithms that can be developed for graphs of the series obtained are of interest. It should also be noted that the proof of Lemma 2 implies the existence of a general analytical method for calculating the distance function in all graphs
of $. This property in turn allowed us to develop a general analytical routing algorithm for all families of $. The routing algorithm is described in [18, 30] and implemented in NoC using the example of one of the possible families of $ as a NoC topology.
Thus, as we could demonstrate on the example of the series $ of circulans, a general structure of graphs of each found series makes it possible to develop a general analytical method for calculating the distance function and then to obtain the shortest-path vectors for a routing algorithm for all graphs included in the series, changing only the value of the parameter p(d), which defines the form of a considered family of circulants.
The second interesting question is whether there are other series of circulant graphs analytically described and given by two parameters d and p(d) that have the same structural and communication properties? Or is it a common property in the space of analytical descriptions of circulant graphs? Are there other designs of circulant series that contain the largest possible graphs for a given degree and a given diameter? Generalizing this problem, we can say that the question is to discover new regularities in obtaining series of circulant graphs that establish relationships between the analytical connection of the order and generators of a graph and the geometry of the resulting series of graphs.
Acknowledgments
The author thanks O. G. Monakhov for useful discussions.
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