Bordered Complex Hadamard Matrices and Strongly Regular Graphs

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.In § 2, a fibre-type construction for graphs is given which, applied to block designs withλ= 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande[22],in which the number of points in the intersection of any pair of blocks attains only two values.

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Uniform Mixing and Association Schemes

The Electronic Journal of Combinatorics ◽

10.37236/4745 ◽

2017 ◽

Vol 24 (3) ◽

Cited By ~ 2

Author(s):

Chris Godsil ◽

Natalie Mullin ◽

Aidan Roy

Keyword(s):

Continuous Time ◽

Hadamard Matrices ◽

Quantum Walks ◽

Association Schemes ◽

Strongly Regular Graphs ◽

Regular Graphs ◽

Paley Graph ◽

Time Quantum ◽

Strongly Regular ◽

Distance Regular Graphs

We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing.First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph $X$ with $n$ vertices, then $n$ is divisible by four. We also prove that if $X$ is bipartite and regular, then $n$ is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on $C_{2m}$ for $m \geq 3$. Using a result of Haagerup, we show that uniform mixing does not occur on $C_p$ for any prime $p$ such that $p \geq 5$. In contrast to this result, we see that $\epsilon$-uniform mixing occurs on $C_p$ for all primes $p$.

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Hadamard Matrices and Strongly Regular Graphs with the $3$-e.c. Adjacency Property

The Electronic Journal of Combinatorics ◽

10.37236/1545 ◽

2000 ◽

Vol 8 (1) ◽

Cited By ~ 1

Author(s):

Anthony Bonato ◽

W. H. Holzmann ◽

Hadi Kharaghani

Keyword(s):

Hadamard Matrix ◽

Infinite Family ◽

Hadamard Matrices ◽

Strongly Regular Graphs ◽

Regular Graphs ◽

Element Subset ◽

Strongly Regular ◽

Almost All ◽

Paley Graphs

A graph is $3$-e.c. if for every $3$-element subset $S$ of the vertices, and for every subset $T$ of $S$, there is a vertex not in $S$ which is joined to every vertex in $T$ and to no vertex in $S\setminus T$. Although almost all graphs are $3$-e.c., the only known examples of strongly regular $3$-e.c. graphs are Paley graphs with at least $29$ vertices. We construct a new infinite family of $3$-e.c. graphs, based on certain Hadamard matrices, that are strongly regular but not Paley graphs. Specifically, we show that Bush-type Hadamard matrices of order $16n^2$ give rise to strongly regular $3$-e.c. graphs, for each odd $n$ for which $4n$ is the order of a Hadamard matrix.

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