Normal matrices and their principal submatrices of co-order one

Savchenko, Sergey Valerievich

doi:10.1016/j.laa.2006.05.018

Cited by 5 publications

(5 citation statements)

References 31 publications

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“…For each index v of A, denote by A − v the matrix of order n − 1 obtained from A by removing its vth row and column. As we have seen above, RLT n is a vertex-transitive tournament and hence, according to [28], for each index v of A, we have…”

Section: On the Number Of 6-cycles In Rlt Nmentioning

confidence: 99%

“…It was also shown in [28] that the one-vertex-deleted subtournament RLT n − v (whose adjacency matrix coincides with A − v) is isomorphic to the so-called Brualdi-Li tournament. (According to the Brualdi-Li conjecture [10] that has been recently confirmed in [13], it maximizes the spectral radius in the class of tournaments of given even order.…”

Section: On the Number Of 6-cycles In Rlt Nmentioning

confidence: 99%

“…Proof Let us first determine

C o r r_{6} (R L T_{n}) .

Since

c_{3} (N^{+} (i)) = c_{3} (N^{-} (i)) = 0

for each vertex i in

R L T_{n}

with

n = 2 δ + 1,

from (25) we have

{italicCorr}_{6} (R L T_{n}, i) = \frac{δ^{2} (δ + 1) (3 δ - 1)}{4} .

Hence,

{italicCorr}_{6} (R L T_{n}) = (2 δ + 1) C o r r_{6} (R L T_{n}, i) = \frac{1}{4} δ^{2} (δ + 1) (2 δ + 1) (3 δ - 1)

\begin{matrix} true \\ right = \frac{1}{64} n {(n - 1)}^{2} (n + 1) (3 n - 5) . \end{matrix}

Let us now determine the trace

t r_{6} (R L T_{n})

of the 6th power of the adjacency matrix A of the tournament

R L T_{n} .

For each index v of A , denote by

A - v

the matrix of order

n - 1

obtained from A by removing its v th row and column. As we have seen above,

R L T_{n}

is a vertex‐transitive tournament and hence, according to , for each index v of A , we have …”

Section: On the Number Of 6‐cycles In Rltnmentioning

confidence: 99%

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On 5‐Cycles and 6‐Cycles in Regular n‐Tournaments

Savchenko

2016

Journal of Graph Theory

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Let T be a tournament of order n and c m (T ) be the number of cycles of length m in T. For m = 3 and odd n, the maximum of c m (T ) is achieved for any regular tournament of order n (M. G. Kendall and B. Babington Smith, 1940), and in the case m = 4, it is attained only for the unique regular locally transitive tournament RLT n of order n (U. Colombo, 1964). A lower bound was also obtained for c 4 (T ) in the class R n of regular tournaments of order n (A. Kotzig, 1968). This bound is attained if and only if T is doubly regular (when n ≡ 3 mod 4) or nearly doubly regular (when n ≡ 1 mod 4) (B. Alspach and C. Tabib, 1982). In the present article, we show that for any regular tournament T of order n, the equality 2c 4 (T ) + c 5 (T ) = n(n − 1)(n + 1)(n − 3)(n + 3)/160 holds. This allows us to reduce the case m = 5 to the case m = 4. In turn, the pure spectral expression for c 6 (T ) obtained in the class R n implies that for a regular tournament T of order n ≥ 7, the inequality c 6 (T ) ≤ n(n − 1)(n + 1)(n − 3)(n 2 − 6n + 3)/384 holds, with equality if and only if T is doubly regular or T is the unique regular tournament of order 7 that is neither doubly regular nor locally transitive. We also determine the value of c 6 (RLT n ) and conjecture that this value coincides with the minimum number of 6-cycles in the class R n for each odd n ≥ 7.

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Section: On the Number Of 6-cycles In Rlt Nmentioning

confidence: 99%

Section: On the Number Of 6-cycles In Rlt Nmentioning

confidence: 99%

“…Proof Let us first determine

C o r r_{6} (R L T_{n}) .

Since

c_{3} (N^{+} (i)) = c_{3} (N^{-} (i)) = 0

for each vertex i in

R L T_{n}

with

n = 2 δ + 1,

from (25) we have

{italicCorr}_{6} (R L T_{n}, i) = \frac{δ^{2} (δ + 1) (3 δ - 1)}{4} .

Hence,

{italicCorr}_{6} (R L T_{n}) = (2 δ + 1) C o r r_{6} (R L T_{n}, i) = \frac{1}{4} δ^{2} (δ + 1) (2 δ + 1) (3 δ - 1)

\begin{matrix} true \\ right = \frac{1}{64} n {(n - 1)}^{2} (n + 1) (3 n - 5) . \end{matrix}

Let us now determine the trace

t r_{6} (R L T_{n})

of the 6th power of the adjacency matrix A of the tournament

R L T_{n} .

For each index v of A , denote by

A - v

the matrix of order

n - 1

obtained from A by removing its v th row and column. As we have seen above,

R L T_{n}

is a vertex‐transitive tournament and hence, according to , for each index v of A , we have …”

Section: On the Number Of 6‐cycles In Rltnmentioning

confidence: 99%

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On 5‐Cycles and 6‐Cycles in Regular n‐Tournaments

Savchenko

2016

Journal of Graph Theory

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“…Since then, the problem has attracted the attention of many researchers, e.g. see [2,10,17,21,23] but, the question for 1 < k < n − 1 remains open. More specifically, it would be interesting to solve the following.…”

Section: Further Researchmentioning

confidence: 99%

Generalized interlacing inequalities

Poon

Sze

2012

Linear and Multilinear Algebra

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“…There is some literature concerning principal (not necessarily normal) submatrices of normal matrices. We mention here one of the earliest papers on that other problem, by Thompson [9], and two of the most recent, one by Malamud [7], and one by Savchenko [8].…”

Section: Introductionmentioning

confidence: 99%