On an infinite sequence of strongly regular digraphs with parameters $(9(2n+3), 3(2n+3), 2n+4, 2n+1, 2n+4)$

On an infinite sequence of strongly regular digraphs with parameters (9(2 n + 3) , 3(2 n + 3) , 2 n + 4 , 2 n + 1 , 2 n + 4) Viktor A. Byzov and Igor A. Pushk arev Marc h, 2026 Abstract The paper constructs an infinite sequence of strongly regular directed graphs. The con- struction is based on representing adjacency matrices as blo c k matrices comp osed of circulant blo c ks, together with the use of a compactification op eration consisten t with polynomial arith- metic modulo x 2 n +3 − 1. Using computer searc h with the p ychoco library and subsequen t anal- ysis of automorphism groups in the GAP system, a stable structural pattern was iden tified, whic h made it p ossible to form ulate and pro ve an explicit form ula for the adjacency matrices of the infinite sequence of directed graphs. Among the obtained digraphs, there are examples with parameters (63 , 21 , 8 , 5 , 8) and (81 , 27 , 10 , 7 , 10), for which the question of existence had previously remained op en. A hypothesis on the structure of the automorphism groups of the digraphs in the constructed sequence is also form ulated. 2020 Mathematics Sub ject Classification: 05C20. Keyw ords: directed strongly regular graph, circulan t matrix, compactification of matrices, automorphism group, p ychoco, GAP . 1 In tro duction This pap er deals with directed graphs (hereafter referred to as digraphs) without lo ops or m ultiple arcs in the same direction. The adjacency matrix of such digraphs is a square matrix with rows and columns indexed by the integers from 1 to n , where n is the num b er of vertices; the en try at the intersection of the i -th row and j -th column is 1 if and only if there is an arc from vertex i to v ertex j . All other entries of the adjacency matrix are zero. W e describ e an infinite family of strongly regular directed graphs with parameters (9(2 n + 3) , 3(2 n + 3) , 2 n + 4 , 2 n + 1 , 2 n + 4) for integer n ⩾ 1. T o construct the digraphs in this sequence, w e use an approach based on representing the adjacency matrix in blo c k-circulant form. This idea, b orro wed from the w ork [1], was also applied b y the authors in the work [2] to find strongly regular directed graphs with parameters (22 , 9 , 6 , 3 , 4). Ho wev er, since the target digraphs in the present article ha ve significan tly more v ertices, several additional constrain ts are imposed on the adjacency matrix to narro w the searc h space. The authors dev elop ed a program that, using the constraint programming library pyc ho co, finds adjacency matrices of dsrg(9(2 n + 3) , 3(2 n + 3) , 2 n + 4 , 2 n + 1 , 2 n + 4) for n = 1 , . . . , 5. F or the obtained digraphs, the automorphism groups were found using the GAP system (see [3]). The digraphs constructed computationally made it p ossible to formulate and prov e a theorem on an infinite sequence of strongly regular digraphs. The pap er is organized as follows. Section 2 introduces the necessary concepts and definitions and presen ts auxiliary statements. Section 3 describ es the metho d of computer-assisted search for strongly regular digraphs. Section 4 prov es the theorem on an infinite sequence of strongly regular digraphs. Viktor A. Byzov, Igor A. Pushk arev: Department of Applied Mathematics and Informatics, Vy atk a State Universit y , Kirov, Russia; e-mail : vbyzo v@yandex.ru, god sha@mail.ru 1 2 Preliminaries The following notation for standard matrices will b e used throughout the text. I v denotes the iden tity matrix of order v , and J v denotes the square matrix of order v with all entries equal to one. The notion of a strongly regular digraph w as first introduced by A. M. Duv al in [4] as a natural generalization of strongly regular undirected graphs. W e now give tw o equiv alent definitions of suc h digraphs. Definition 1. A strongly regular digraph with parameter set ( v , k , t, λ, µ ) is a directed graph on v vertices satisfying the following conditions: 1. The outdegree and indegree of eac h v ertex are both equal to k ; 2. F or each vertex x , there are exactly t paths of the form x → z → x ; 3. If there is a directed edge from x to y  = x , then there exist exactly λ paths of the form x → z → y ; 4. If there is no directed edge from x to y  = x , then there exist exactly µ paths of the form x → z → y . Definition 2. A strongly regular digraph with parameter set ( v , k , t, λ, µ ) is a directed graph on v vertices whose adjacency matrix A satisfies the following conditions: A 2 = tI v + λA + µ ( J v − I v − A ) , AJ v = J v A = k J v . Instead of the phrase “strongly regular digraph with parameter set ( v , k , t, λ, µ )”, we will often use the follo wing shorter notation in the text: dsrg( v , k , t, λ, µ ). A circulant matrix (or circulant) is defined as a square matrix in which eac h row, starting from the second, is obtained b y a cyclic righ t shift of the preceding row by one p osition. Denote b y Circ Z ( n ) the ring of circulant matrices of order n with entries from Z . It is known that the ring Circ Z ( n ) is isomorphic to the quotien t ring Z [ x ]  ( x n − 1) consisting of p olynomials of degree less than n (see, for example, [5]). Under this isomorphism, the matrix      a 1 a 2 . . . a n a n a 1 . . . a n − 1 . . . . . . . . . . . . a 2 a 3 . . . a 1      (1) corresp onds to the p olynomial a 1 + a 2 x + . . . + a n x n − 1 . Let M b e a blo ck matrix whose blo cks are all circulant matrices. The compactification of the matrix M is defined as the matrix M ( x ) obtained b y replacing each blo c k of M with the corresp onding p olynomial under the isomorphism describ ed ab ov e. As an example, consider one of the digraphs dsrg(8 , 3 , 2 , 1 , 1), whose adjacency matrix can be represen ted as four circulant blo c ks of order four: S =             0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0             . (2) The matrix S can be compactified to the matrix S ( x ) =  x 3 x + x 2 x 2 + x 3 x  . (3) 2 The inv erse op eration, which transforms a compactified matrix into a binary matrix consisting of circulant blo c ks, will b e called decompactification. A key prop ert y of compactification is that this operation is compatible with matrix addition and m ultiplication. Sp ecifically , let M 1 and M 2 b e tw o square blo ck matrices consisting of circulant blo c ks of order n , and let M 3 = M 1 + M 2 , M 4 = M 1 · M 2 . Then M 3 ( x ) ≡ M 1 ( x ) + M 2 ( x ) (mo d x n − 1), M 4 ( x ) ≡ M 1 ( x ) · M 2 ( x ) (mo d x n − 1), where M 1 ( x ), M 2 ( x ), M 3 ( x ), and M 4 ( x ) are the compactifications of the corresp onding matrices. W e use the notation M [ i, j ] for the entry of matrix M located at the intersection of the i -th ro w and j -th column, where ro ws and columns are n umbered starting from one. 3 A computer exp erimen t to find strongly regular digraphs W e seek the adjacency matrices A n of dsrg(9(2 n + 3) , 3(2 n + 3) , 2 n + 4 , 2 n + 1 , 2 n + 4) in the form of 9 × 9 blo c k matrices, where each blo c k is a circulan t matrix of order 2 n + 3. F rom Definition 2, it follows that the matrix A n satisfies the conditions A 2 n + 3 A n = (2 n + 4) J 9(2 n +3) , (4) A n J 9(2 n +3) = J 9(2 n +3) A n = 3(2 n + 3) J 9(2 n +3) . (5) Denote by A n ( x ) the compactification of the matrix A n , and b y Q n ( x ) = 2 n +2 P i =0 x i the p olyno- mial corresp onding to the all-ones circulan t block. Then the matrix A n ( x ) satisfies the following congruences: A n ( x ) 2 + 3 A n ( x ) ≡ (2 n + 4) J 9 Q n ( x ) (mo d x 2 n +3 − 1) , (6) A n ( x ) J 9 Q n ( x ) ≡ J 9 A n ( x ) Q n ( x ) ≡ 3(2 n + 3) J 9 Q n ( x ) (mo d x 2 n +3 − 1) . (7) Since x = 1 is a ro ot of the polynomial x 2 n +3 − 1, substituting into congruences (6) and (7) yields the equalities A n (1) 2 + 3 A n (1) = (2 n + 3)(2 n + 4) J 9 , (8) A n (1) J 9 = J 9 A n (1) = 3(2 n + 3) J 9 . (9) The following lemma describ es a sequence of matrices that satisfy equations (8) and (9). Lemma 3. The se quenc e of matric es C n =               0 n + 1 n + 1 n + 1 n + 1 1 2 n + 1 n + 1 0 n + 1 n + 1 n + 1 n + 1 1 2 n + 1 n + 1 0 n + 1 n + 1 n + 1 n + 1 1 2 n + 1 n + 1 0 n + 1 n + 1 n + 1 n + 1 1 2 n + 1 n + 1 0 n + 1 n + 1 n + 1 n + 1 1 2 n + 1 n + 1 0 n + 1 n + 1 n + 1 n + 1 1 2 n + 1 n + 1 2 n + 3 1 1 1 1 2 n + 1 2 n − 1 1 1 2 n + 3 1 1 1 1 2 n + 1 2 n − 1 1 1 2 n + 3 1 1 1 1 2 n + 1 2 n − 1 1 1               (10) has the fol lowing pr op erties: C 2 n + 3 C n = (2 n + 3)(2 n + 4) J 9 , (11) C n J 9 = J 9 C n = 3(2 n + 3) J 9 . (12) The v alidity of this lemma can b e verified by direct substitution of the formula for C n in to equations (11) and (12). The verification can b e p erformed by direct computation or using a computer algebra system. Th us, the sequence of matrices C n can b e regarded as a “p oten tial candidate” for the role of A n (1). 3 R emark 4 . Note that a suitable sequence of matrices C n w as obtained through nonsystematic computer exp eriments using the constraint programming library pyc ho co (see [6]). W e will search sp ecifically for the matrices A n ( x ), since the adjacency matrices of the digraphs can b e uniquely recov ered from them. Exhaustive enumeration of all p ossible A n ( x ) is infeasible ev en for small n , so we imp ose the follo wing additional constraints to reduce the searc h space. 1. A n (1) = C n . 2. A n ( x )[1 , 2] = 1 + x + x 2 + . . . + x n . 3. A n ( x )[7 , 2] = 1. 4. F or 1 ⩽ j ⩽ 9, A n ( x )[7 , j ] = A n ( x )[8 , j ] = A n ( x )[9 , j ] . (13) 5. F or 2 ⩽ i ⩽ 6, 1 ⩽ j ⩽ 9, A n ( x )[ i, j ] ≡ x · A n ( x )[ i − 1 , j ] (mo d x 2 n +3 − 1) . (14) 6. F or 1 ⩽ i ⩽ 9, A n ( x )[ i, 3] ≡ x · A n ( x )[ i, 2] (mo d x 2 n +3 − 1) . (15) 7. F or 1 ⩽ i ⩽ 9, A n ( x )[ i, 8] ≡ x · A n ( x )[ i, 9] (mo d x 2 n +3 − 1) . (16) R emark 5 . The describ ed conditions w ere “guessed” through n umerous computer exp erimen ts: with this set of conditions, the softw are search describ ed b elow yielded p ositive results. A program was written to search for matrices A n ( x ) satisfying conditions (1)–(7). The con- strain t programming library p yc ho co was used. The program found all suitable matrices for n = 1 , . . . , 5. F or the found digraphs, the automorphism groups were computed using the GAP system. The results are presented in T able 1. The third column of this table giv es the num b er of nonisomorphic digraphs among those found. T able 1: Searc h results for strongly regular digraphs n P arameters Searc h time (sec.) Num b er of dsrgs Automorphism groups 1 (45 , 15 , 6 , 3 , 6) 13.85 1 C 2 × ( C 4 2 ⋊ C 5 ) 3 C 2 2 × ( C 8 2 ⋊ C 5 ) 2 C 2 × ( C 5 3 ⋊ ( C 2 × ( C 8 2 ⋊ C 5 ))) 2 (63 , 21 , 8 , 5 , 8) 62.10 4 C 2 × ( C 6 2 ⋊ C 7 ) 3 (81 , 27 , 10 , 7 , 10) 3388 2 C 2 × ( C 8 2 ⋊ C 9 ) 4 (99 , 33 , 12 , 9 , 12) 1703 2 C 2 × ( C 10 2 ⋊ C 11 ) 5 (117 , 39 , 14 , 11 , 14) 90354 2 C 2 × ( C 12 2 ⋊ C 13 ) The searc h was p erformed on a computer with an In tel Core i5-7400 pro cessor (3.0 GHz) and 32 GB of RAM. All found digraphs are av ailable in the rep ository [7]. Note that, at the time of writing, there was no information in T able [8] regarding the existence of digraphs with parameters (63 , 21 , 8 , 5 , 8) and (81 , 27 , 10 , 7 , 10). 4 4 An infinite sequence of strongly regular digraphs It was observ ed that several of the found digraphs hav e similar structures. This led to a conjecture whose pro of yielded an infinite sequence of strongly regular digraphs. In tro duce the following notation. Let P ( x ) = n P i =0 x i , Q ( x ) = 2 n +2 P i =0 x i , R ( x ) = Q ( x ) − x n − 1 − x 2 n +1 , S ( x ) = Q ( x ) − 1 − x 2 − x n +2 − x n +3 . W e now prov e some useful prop erties of these p olynomials. Lemma 6. The fol lowing statements hold (al l c ongruenc es ar e mo dulo x 2 n +3 − 1 ): 1. x s Q ( x ) ≡ Q ( x ) for any inte ger s ⩾ 0 ; 2. P ( x ) · Q ( x ) ≡ ( n + 1) Q ( x ) ; 3. Q ( x ) 2 ≡ (2 n + 3) Q ( x ) ; 4. (1 − x ) P ( x ) = 1 − x n +1 , (1 − x ) Q ( x ) = 1 − x 2 n +3 . Pr o of. The v alidity of item (1) follows, for example, from the fact that multiplication by x s simply cyclically shifts the exp onen ts, which leav es Q ( x ) unchanged since this p olynomial contains all p ossible exp onents from 0 to 2 n + 2. The v alidit y of items (2) and (3) follows from item (1) and the decomp ositions P ( x ) · Q ( x ) = P n i =0 x i Q ( x ) and Q ( x ) 2 = P 2 n +2 i =0 x i Q ( x ). The last item of the lemma is verified by direct computation. The lemma is prov ed. Consider the sequence of matrices A n ( x ) for n ⩾ 2, defined by the formula               0 P xP x n − 1 P x n − 2 P x 2 n x (1 + x n ) x 2 P xP 0 xP x 2 P x n P x n − 1 P x 2 n +1 x 2 (1 + x n ) x 3 P x 2 P 0 x 2 P x 3 P x n +1 P x n P x 2 n +2 x 3 (1 + x n ) x 4 P x 3 P 0 x 3 P x 4 P x n +2 P x n +1 P 1 x 4 (1 + x n ) x 5 P x 4 P 0 x 4 P x 5 P x n +3 P x n +2 P x x 5 (1 + x n ) x 6 P x 5 P 0 x 5 P x 6 P x n +4 P x n +3 P x 2 x 6 (1 + x n ) x 7 P x 6 P Q 1 x x n − 1 x n − 2 R S x 2 x Q 1 x x n − 1 x n − 2 R S x 2 x Q 1 x x n − 1 x n − 2 R S x 2 x               , (17) where all op erations are p erformed in the ring Z [ x ]  ( x 2 n +3 − 1). F or brevity , P ( x ), Q ( x ), R ( x ), and S ( x ) are denoted simply by P , Q , R , and S , resp ectively . Theorem 7. The digr aph whose adjac ency matrix c oincides with the de c omp actific ation of ma- trix (17) for n ⩾ 2 is dsr g (9(2 n + 3) , 3(2 n + 3) , 2 n + 4 , 2 n + 1 , 2 n + 4) . Pr o of. T o prov e the theorem, it suffices to show that the sequence of matrices A n ( x ) for n ⩾ 2 satisfies the regularity condition and congruence (6). The fact that A n ( x ) satisfies the regularit y condition follows from Lemma 3 and A n (1) = C n . W e now prov e that A n ( x ) 2 + 3 A n ( x ) ≡ (2 n + 4) J 9 Q ( x ). Denote A n ( x ) 2 + 3 A n ( x ) by W ( x ). Since in the first six ro ws of A n ( x ), eac h row starting from the second is obtained by multiplying the previous row by x , it suffices to compute only the first ro w of W ( x ). Similarly , due to the equalit y of the last three rows of A n ( x ), we need to compute only the seven th row instead of the last three ro ws of W ( x ). In the computations, we will use the prop erties from Lemma 6. W ( x )[1 , 1] = ( x 2 P ( x ) + xP ( x ) + x n +1 + x ) Q ( x ) ≡ ≡ ( n + 1) Q ( x ) + ( n + 1) Q ( x ) + 2 Q ( x ) = (2 n + 4) Q ( x ) . (18) W e now pro ceed to compute the next entry of the matrix W ( x ): W ( x )[1 , 2] = (2 x n +2 + x 3 + x ) P ( x ) 2 + ( x 2 n +5 + x 2 + x + 3) P ( x ) + x n +1 + x. (19) 5 Using Lemma 6 and the relation x 2 n +3 ≡ 1, it can b e shown that (1 − x ) W ( x )[1 , 2] ≡ 0 (mo d x 2 n +3 − 1) . (20) Hence, there exists a p olynomial T ( x ) in the ring Z [ x ] suc h that (1 − x ) W ( x )[1 , 2] = T ( x )(1 − x 2 n +3 ) = T ( x )(1 − x ) Q ( x ) . (21) Since Z [ x ] is an integral domain and 1 − x  = 0, equation (21) implies that W ( x )[1 , 2] = T ( x ) Q ( x ) . (22) Since deg W ( x )[1 , 2] ⩽ 2 n + 2 and deg Q ( x ) = 2 n + 2, the p olynomial T ( x ) must b e a constant. As W (1)[1 , 2] = 2(2 n + 3)( n + 2) and Q (1) = 2 n + 3, it follows that T ( x ) = 2 n + 4. Th us, W ( x )[1 , 2] = (2 n + 4) Q ( x ). F urthermore, W ( x )[1 , 3] = x · W ( x )[1 , 2] ≡ (2 n + 4) Q ( x ). W ( x )[1 , 4] = x n − 1 · W ( x )[1 , 2] ≡ (2 n + 4) Q ( x ). W ( x )[1 , 5] = x n − 2 · W ( x )[1 , 2] ≡ (2 n + 4) Q ( x ). After simplification, w e obtain W ( x )[1 , 6] ≡ (2 n + 4) Q ( x ) + (1 − x ) P ( x )( x 2 n +1 + x n + 2 x n − 1 )+ + 2 x 2 n − x n − x n − 1 . (23) Item (4) of Lemma 6 allows us to simplify: W ( x )[1 , 6] ≡ (2 n + 4) Q ( x ). Using the expression for S ( x ), we obtain W ( x )[1 , 7] ≡ ≡ (2 n + 4) Q ( x ) + (1 − x ) P ( x )( x n +4 + 2 x n +3 + x n +2 + x 2 + x + 2) − − x n +4 − x n +3 + 2 x n +1 + x − 1 . (24) Item (4) of Lemma 6 allows the simplification: W ( x )[1 , 7] ≡ (2 n + 4) Q ( x ). F urthermore, W ( x )[1 , 8] = x 2 · W ( x )[1 , 2] ≡ (2 n + 4) Q ( x ). W ( x )[1 , 9] = x · W ( x )[1 , 2] ≡ (2 n + 4) Q ( x ). W ( x )[7 , 1] = ( S ( x ) + x 2 + x + 3) Q ( x ) = = ( Q ( x ) − 1 − x n +2 − x n +3 + x + 3) Q ( x ) ≡ (2 n + 4) Q ( x ) . (25) Using the statemen ts of Lemma 6, w e obtain W ( x )[7 , 2] ≡ (2 n + 3) Q ( x ) + x n +2 P ( x ) + xP ( x ) + 1 . (26) F rom the definitions of P ( x ) and Q ( x ), it follows that x n +2 P ( x ) + xP ( x ) + 1 = Q ( x ), so W ( x )[7 , 2] ≡ (2 n + 4) Q ( x ). F urthermore, W ( x )[7 , 3] = x · W ( x )[7 , 2] ≡ (2 n + 4) Q ( x ). W ( x )[7 , 4] = x n − 1 · W ( x )[7 , 2] ≡ (2 n + 4) Q ( x ). W ( x )[7 , 5] = x n − 2 · W ( x )[7 , 2] ≡ (2 n + 4) Q ( x ). Using items (1) and (3) of Lemma 6 and a sequence of transformations, it is prov ed that W ( x )[7 , 6] ≡ W ( x )[7 , 7] ≡ (2 n + 4) Q ( x ). F urthermore, W ( x )[7 , 8] = x 2 · W ( x )[7 , 2] ≡ (2 n + 4) Q ( x ). W ( x )[7 , 9] = x · W ( x )[7 , 2] ≡ (2 n + 4) Q ( x ). Th us, we hav e sho wn that all entries in the first and seven th ro ws of the matrix W ( x ) equal (2 n + 4) Q ( x ). Consequently , all other entries of this matrix also equal (2 n + 4) Q ( x ). The theorem is prov ed. The authors developed a program to generate strongly regular digraphs based on the formula from Theorem 7. F or the generated digraphs, the automorphism groups w ere computed using GAP , which led to the follo wing conjecture. Conjecture 8. F or the constructed sequence of strongly regular digraphs with parameters (9(2 n + 3) , 3(2 n + 3) , 2 n + 4 , 2 n + 1 , 2 n + 4), the automorphism group is C 2 × ( C 2 n +2 2 ⋊ C 2 n +3 ). 6 5 Conclusion In this paper, an explicit form ula is obtained for a family of strongly regular digraphs with parame- ters (9(2 n + 3) , 3(2 n + 3) , 2 n + 4 , 2 n + 1 , 2 n + 4). The adjacency matrix of each digraph consists of 81 circulan t submatrices. A conjecture is form ulated that the automorphism groups of the obtained digraphs are C 2 × ( C 2 n +2 2 ⋊ C 2 n +3 ). References [1] O. Gritsenko On strongly regular graph with parameters (65; 32; 15; 16). 2021. 10 p. (Cornell Univ. Libr. e-Prin t Archiv e; arXiv:2102.05432). doi:10.48550/arXiv.2102.05432. [2] V. A. Byzov and I. A. 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