Geodesic
A characterization of graphs by codes from their incidence matrices
Peter Dankelmann1 ; Jennifer D. Key2 ; Bernardo G. Rodrigues3
1University of Johannesburg
2Clemson University
3University of KwaZulu-Natal
The electronic journal of combinatorics, Tome 20 (2013) no. 3
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We continue our earlier investigation of properties of linear codes generated by the rows of incidence matrices of $k$-regular connected graphs on $n$ vertices. The notion of edge connectivity is used to show that, for a wide range of such graphs, the $p$-ary code, for all primes $p$, from an $n \times \frac{1}{2}nk$ incidence matrix has dimension $n$ or $n-1$, minimum weight $k$, the minimum words are the scalar multiples of the rows, there is a gap in the weight enumerator between $k$ and $2k-2$, and the words of weight $2k-2$ are the scalar multiples of the differences of intersecting rows of the matrix. For such graphs, the graph can thus be retrieved from the code.
DOI : 10.37236/2770
Classification : 05C50, 05B20, 05C40, 94B05
Mots-clés : linear codes, connected graphs, edge-connectivity, incidence matrix

Peter Dankelmann  1   ; Jennifer D. Key  2   ; Bernardo G. Rodrigues  3

1 University of Johannesburg
2 Clemson University
3 University of KwaZulu-Natal 
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Peter Dankelmann; Jennifer D. Key; Bernardo G. Rodrigues. A characterization of graphs by codes from their incidence matrices. The electronic journal of combinatorics, Tome 20 (2013) no. 3. doi: 10.37236/2770

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