English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

In this paper, we study a conjecture of Andries E. Brouwer from 1996regarding the minimum number of vertices of a strongly regular graph whoseremoval disconnects the graph into non-singleton components.  We show that strongly regular graphs constructed from copolar spaces and fromthe more general spaces called $\Delta$-spaces are counterexamples to Brouwer'sConjecture. Using J.I. Hall's characterization of finite reduced copolarspaces, we find that the triangular graphs $T(m)$, the symplectic graphs$Sp(2r,q)$ over the field $\mathbb{F}_q$ (for any $q$ prime power), and thestrongly regular graphs constructed from the hyperbolic quadrics $O^{+}(2r,2)$and from the elliptic quadrics $O^{-}(2r,2)$ over the field $\mathbb{F}_2$,respectively, are counterexamples to Brouwer's Conjecture. For each of thesegraphs, we determine precisely the minimum number of vertices whose removaldisconnects the graph into non-singleton components. While we are not aware ofan analogue of Hall's characterization theorem for $\Delta$-spaces, we showthat complements of the point graphs of certain finite generalized quadranglesare point graphs of $\Delta$-spaces and thus, yield other counterexamples toBrouwer's Conjecture.  We prove that Brouwer's Conjecture is true for many families of stronglyregular graphs including the conference graphs, the generalized quadrangles$GQ(q,q)$ graphs, the lattice graphs, the Latin square graphs, the stronglyregular graphs with smallest eigenvalue -2 (except the triangular graphs) andthe primitive strongly regular graphs with at most 30 vertices except for fewcases.  We leave as an open problem determining the best general lower bound for theminimum size of a disconnecting set of vertices of a strongly regular graph,whose removal disconnects the graph into non-singleton components.

On a conjecture of Brouwer involving the connectivity of strongly regular graphs