Abstract
Spectral graph theory uses the eigenvalues of matrices associated with a graph to determine the structural properties of the graph. The spectrum of the generalized adjacency matrix is considered in the paper. Graphs with the same spectrum are called cospectral. Is every graph uniquely determined by its spectrum (DS for short)? This question goes back for about half a century, and originates from chemistry. In 1956 Gunthard and Primas raised the question in a paper that related the theory of graph spectra to Huckel’s theory. At that time it was believed that every graph is determined by the spectrum, until in 1957 Collatz and Sinogowitz presented a pair of cospectral trees. In 1967 Schwenk proved that for almost all trees there is another tree with the same spectrum. Such a statement is neither proved nor refuted for the class of graphs in general. Till now, computational experiments were done on the set of all graphs on up to 12 vertices by Haemers. Computer enumerations for small n show that up to 10 vertices the fraction of graphs that are DS decreases, but for n = 11 and n = 12 it increases again. We consider the construction of the cospectral graphs called GM-switching for graph G taking the cycle C2n and adjoining a vertex v adjacent to half the vertices of C2n. For these graphs we determine the pairs of cospectral nonisomorphic graphs for small n. It is an operation on graphs that leaves the spectrum of the generalized adjacency matrix invariant. It turns out that for the enumerated cases a large part of all cospectral graphs comes from GM switching, and that the fraction of graphs on n vertices with a cospectral mate starts to decrease at some value of n < 11 (depending on the matrix). Since the fraction of cospectral graphs on n vertices constructible by GM switching tends to 0 if n → ∞, the present data give some indication that possibly almost no graph has a cospectral mate. Haemers and Spence derived asymptotic lower bounds for the number of graphs with a cospectral mate from GM switching.
Highlights
Спектральна теорiя графiв — напрям у теорiї графiв, що вивчає спектральнi властивостi матриць, асоцiйованих iз графами
Якщо матрицi M та Mє матрицями сумiжностi графiв, то GM-комутацiя дозволяє побудувати коспектральнi доповнення
For these graphs we determine the pairs of cospectral nonisomorphic graphs for small n
Summary
Спектральна теорiя графiв використовує власнi значення матриць, асоцiйованих iз графом, для визначення структурних властивостей графа. У статтi розглянуто спектр узагальненої матрицi сумiжностi. Що графи, у яких менше, нiж п’ять вершин, однозначно визначаються своїм спектром (будемо називати їх DS — вiд англiйського «determined by its spectrum») вiдносно звичайної матрицi сумiжностi. Для графiв на шiстьох вершинах iснує п’ять пар коспектральних графiв, зображених на рис. 3. Проте жоден iз графiв, побудованих на шести i менше вершинах, не має коспектральної пари вiдносно узагальненої матрицi сумiжностi. У наступному роздiлi введено означення узагальненої матрицi сумiжностi та розглянуто умови, за яких два графи будуть коспектральними вiдносно такої матрицi. 3. Пари коспектральних графiв на 6 вершинах вiдносно A їх кiлькiсть дозволяє побудувати метод GMкомутацiї, який ми розглядаємо далi. У загальному для всiх графiв гiпотеза про те, що майже всi графи є DS графами, залишається не доведеною
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