In this paper, we discuss maximality of Seidel matrices with a fixed largest eigenvalue. We present a classification of maximal Seidel matrices of largest eigenvalue $3$, which gives a classification of maximal equiangular lines in a Euclidean space
with angle $arccos1/3$. Motivated by the maximality of the exceptional root system $E_8$, we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal.
In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s,
t$, we determine the complementary Ramsey numbers $bar{R}(m,t,s)$ for $(s,t)=(4,4)$ and $(3,6)$.
In this paper, we study the characteristic polynomials of the line graphs of generalized Bethe trees. We give an infinite family of such graphs sharing the same smallest eigenvalue. Our family generalizes the family of coronas of complete graphs discovered by Cvetkovic and Stevanovic.
In this paper, we give a combinatorial characterization of the special graphs of fat Hoffman graphs containing $mathfrak{K}_{1,2}$ with smallest eigenvalue greater than -3, where $mathfrak{K}_{1,2}$ is the Hoffman graph having one slim vertex and two fat vertices.
In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least -1-tau, where tau is the golden ratio, can be described by a finite set of fat (-1-tau)-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean t
hat every fat Hoffman graph with smallest eigenvalue at least -1-tau is an H-line graph, where H is the set of isomorphism classes of maximal fat (-1-tau)-irreducible Hoffman graphs. It turns out that there are 37 fat (-1-tau)-irreducible Hoffman graphs, up to isomorphism.
We investigate fat Hoffman graphs with smallest eigenvalue at least -3, using their special graphs. We show that the special graph S(H) of an indecomposable fat Hoffman graph H is represented by the standard lattice or an irreducible root lattice. Mo
reover, we show that if the special graph admits an integral representation, that is, the lattice spanned by it is not an exceptional root lattice, then the special graph S(H) is isomorphic to one of the Dynkin graphs A_n, D_n, or extended Dynkin graphs A_n or D_n.
