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Register a new userThere does not exist a strongly regular graph with parameters $(1911,270,105,27)$
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In this paper we show that there does not exist a strongly regular graph with parameters $(1911,270,105,27)$.
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We consider orbit partitions of groups of automorphisms for the symplectic graph and apply Godsil-McKay switching. As a result, we find four families of strongly regular graphs with the same parameters as the symplectic graphs, including the one discovered by Abiad and Haemers. Also, we prove that switched graphs are non-isomorphic to each other by considering the number of common neighbors of three vertices.
A symplectically invariant definition of special Kahler geometry is discussed. Certain aspects hereof are illustrated by means of Calabi-Yau moduli spaces.
A known Kronecker construction of completely regular codes has been investigated taking different alphabets in the component codes. This approach is also connected with lifting constructions of completely regular codes. We obtain several classes of completely regular codes with different parameters, but identical intersection array. Given a prime power $q$ and any two natural numbers $a,b$, we construct completely transitive codes over different fields with covering radius $rho=min{a,b}$ and identical intersection array, specifically, one code over $F_{q^r}$ for each divisor $r$ of $a$ or $b$. As a corollary, for any prime power $q$, we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters. Under the same conditions, an explicit construction of an infinite family of $q$-ary uniformly packed codes (in the wide sense) with covering radius $rho$, which are not completely regular, is also given.
A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs on $G$ arise from orienting $G$ and then labelling the edges of $G$ with elements of a group $Gamma$. In this case, we may define a biased graph by declaring a cycle to be balanced if the product of the labels on its edges is the identity, with the convention that we take the inverse value for an edge traversed backwards. Our first result gives a natural topological characterisation of biased graphs arising from group-labellings. In the second part of this article, we use this theorem to construct some exceptional biased graphs. Notably, we prove that for every $m ge 3$ and $ell$ there exists a minor minimal not group labellable biased graph on $m$ vertices where every pair of vertices is joined by at least $ell$ edges. Finally, we show that these results extend to give infinite families of excluded minors for certain families of frame and lift matroids.
The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian matrices and hence the maximum of Laplacian spreads among all simple connected graphs of fixed order, minimum degree, maximum degree, minimum size of common neighbors of two adjacent vertices and minimum size of common neighbors of two nonadjacent vertices. Some other extremal graphs are also provided.
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