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Determinants of adjacency matrices of graphs | ||
Transactions on Combinatorics | ||
مقاله 2، دوره 1، شماره 4، اسفند 2012، صفحه 9-16 اصل مقاله (458.28 K) | ||
نوع مقاله: Research Paper | ||
شناسه دیجیتال (DOI): 10.22108/toc.2012.2041 | ||
نویسنده | ||
Alireza Abdollahi* | ||
University of Isfahan | ||
چکیده | ||
We study the set of all determinants of adjacency matrices of graphs with a given number of vertices. Using Brendan McKay's data base of small graphs, determinants of graphs with at most $9$ vertices are computed so that the number of non-isomorphic graphs with given vertices whose determinants are all equal to a number is exhibited in a table. Using an idea of M. Newman, it is proved that if $G$ is a graph with $n$ vertices, $m$ edges and $\{d_1,\dots,d_n\}$ is the set of vertex degrees of $G$, then $\gcd(2m,d^2)$ divides the determinant of the adjacency matrix of $G$, where $d=\gcd(d_1,\dots,d_n)$. Possible determinants of adjacency matrices of graphs with exactly two cycles are obtained. | ||
کلیدواژهها | ||
Determinant؛ adjacency matrices of graphs؛ maximum determinant | ||
مراجع | ||
S. Akbari, E. Ghorbani, H. Kharaghani, and G. B. Khosrovshahi Graphs with extremal determinants preprint
K. P. Costello and V. H. Vu (2008) The rank of random graphs Random Structures Algorithms 33, 269-285
D. M. Cvetkovi\'c, M. Doob and H. Sachs (1980) Theory and Application, Pure and Applied Mathematics Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London 87
S. Fallat and P. van den Driessche (1997) Maximum determinant of $(0,1)$ matrices with certain constant row and column sums Linear and Multilinear Algebra 42 (4), 303-318
W. H. Haemers and Q. Xiang (2010) Strongly regular graphs with parameters $(4m^4, 2m^4 + m^2,m^4 + m^2,m^4 + m^2)$ exist for all $m > 1$ European J. Combin. 31 (6), 1553-1559
J. Hadamard (1968) R\'esolution d'une question relative aux determinants Selecta, pp. 136-142, Gauthier-Villars, Paris, 1935 and Oeuvres, Tome I, CNRS, Paris , 239-245
S. Hu (2003) The Classification and maximum determinants of the adjacency matrices of graphs with one cycle J. Math. Study 36 (1), 102-104
M. Newman (1978) Combinatorial matrices with small determinants Canad. J. Math. 30, 756-762
H. J. Ryser (1956) Maximal determinants in combinatorial investigations Canad. J. Math. 8, 245-249
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