8 papers · 1 filter
Graphs with three eigenvalues from group divisible designs
Gary Greaves, Jose Yip
We construct biregular graphs with three distinct eigenvalues from semiregular group divisible designs. For every prime power , we use the spectral structure of Muzychuk's Const…
Quantum entropy and the Bannai--Ito multiplicity conjecture
Gary Greaves, Haoran Zhu
We prove that the multiplicities of every symmetric -polynomial association scheme form a log-concave sequence. This resolves the 1984 unimodality conjecture of Bannai and Ito a…
Neumaier graphs of coherent rank five
Gary Greaves, Zhao Kuang Tan
We construct an infinite family of Neumaier graphs of coherent rank five, answering the existence question at the smallest possible coherent rank beyond the strongly regular case.…
Characteristic polynomials of -matrices modulo a power of
Gary Greaves, Huu An Phan
For a fixed integer and large enough, we show that the number of congruence classes modulo of characteristic polynomials of symmetric $\{\pm…
A note on some spectral properties of generalised pancake graphs
Gary R. W. Greaves, Haoran Zhu
We prove that the spectral gap of generalised pancake graphs is strictly less than 2 and strictly less than 1 for burnt pancake graphs. In addition, we establish lower bounds on th…
Neumaier graphs from cyclotomy with small coherent rank
Gary R. W. Greaves, Zhao Kuang Tan
Using cyclotomy, we construct a new infinite family of Neumaier graphs that includes infinitely many strongly regular graphs. Notably, this family conjecturally contains infinitely…
The coherent rank of a graph with three eigenvalues
Gary Greaves, Jose Yip
We characterise graphs that have three distinct eigenvalues and coherent ranks 8 and 9, linking the former to certain symmetric 2-designs and the latter to specific quasi-symmetric…
Constructions of -designs from weighing matrices and association schemes
Gary Greaves, Sho Suda
We provide a method to construct -designs from weighing matrices and association schemes. One instance of our method can produce a -design from any (symmetric or skew-symmetr…