Geodesic
Hadamard matrices and strongly regular graphs with the 3-e. c. adjacency property
The electronic journal of combinatorics, Tome 8 (2001) no. 1
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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.
DOI : 10.37236/1545
Classification : 05C50, 05B20, 05E30
Mots-clés : \(n\)-e.c. graphs, Bush-type Hadamard matrix, design 
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     author = {Anthony Bonato and W. H. Holzmann and Hadi Kharaghani},
     title = {Hadamard matrices and strongly regular graphs with the 3-e. c. adjacency property},
     journal = {The electronic journal of combinatorics},
     year = {2001},
     volume = {8},
     number = {1},
     doi = {10.37236/1545},
     zbl = {0955.05068},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1545/}
}
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Anthony Bonato; W. H. Holzmann; Hadi Kharaghani. Hadamard matrices and strongly regular graphs with the 3-e. c. adjacency property. The electronic journal of combinatorics, Tome 8 (2001) no. 1. doi: 10.37236/1545

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