In this paper we show that there does not exist a strongly regular graph with parameters $(1911,270,105,27)$.
In this paper we give two characterizations of the $p times q$-grid graphs as co-edge-regular graphs with four distinct eigenvalues.
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.
Let $Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $Gamma$. Fix a base vertex $xin X$ and for $0 leq i leq D$ let $E^*_i=E^*_i(x)$ denote the projection matrix to th
e $i$th subconstituent space of $Gamma$ with respect to $x$. The Terwilliger algebra $T(x)$ of $Gamma$ with respect to $x$ is the semisimple subalgebra of $mathrm{Mat}_X(mathbb{C})$ generated by $A, E^*_0, E^*_1, ldots, E^*_D$. Remark that the isomorphism class of $T(x)$ depends on the choice of the base vertex $x$. We say $Gamma$ is pseudo-vertex-transitive whenever for any vertices $x,y in X$, the Terwilliger algebras $T(x)$ and $T(y)$ are isomorphic. In this paper we discuss pseudo-vertex transitivity for distance-regular graphs with diameter $Din {2,3,4}$. In the case of diameter $2$, a strongly regular graph $Gamma$ is thin, and $Gamma$ is pseudo-vertex-transitive if and only if every local graph of $Gamma$ has the same spectrum. In the case of diameter $3$, Taylor graphs are thin and pseudo-vertex-transitive. In the case of diameter $4$, antipodal tight graphs are thin and pseudo-vertex-transitive.
In this paper, we study the order of a maximal clique in an amply regular graph with a fixed smallest eigenvalue by considering a vertex that is adjacent to some (but not all) vertices of the maximal clique. As a consequence, we show that if a strong
ly regular graph contains a Delsarte clique, then the parameter $mu$ is either small or large. Furthermore, we obtain a cubic polynomial that assures that a maximal clique in an amply regular graph is either small or large (under certain assumptions). Combining this cubic polynomial with the claw-bound, we rule out an infinite family of feasible parameters $(v,k,lambda,mu)$ for strongly regular graphs. Lastly, we provide tables of parameters $(v,k,lambda,mu)$ for nonexistent strongly regular graphs with smallest eigenvalue $-4, -5, -6$ or $-7$.
Let $Gamma$ be a $Q$-polynomial distance-regular graph of diameter $dgeq 3$. Fix a vertex $gamma$ of $Gamma$ and consider the subgraph induced on the union of the last two subconstituents of $Gamma$ with respect to $gamma$. We prove that this subgraph is connected.
A regular graph is co-edge regular if there exists a constant $mu$ such that any two distinct and non-adjacent vertices have exactly $mu$ common neighbors. In this paper, we show that for integers $sge 2$ and $n$ large enough, any co-edge-regular gra
ph which is cospectral with the $s$-clique extension of the triangular graph $T((n)$ is exactly the $s$-clique extension of the triangular graph $T(n)$.
Signed graphs are graphs whose edges get a sign $+1$ or $-1$ (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention fro
m graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs. Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs.
In this paper we show that for integers $sgeq2$, $tgeq1$, any co-edge-regular graph which is cospectral with the $s$-clique extension of the $ttimes t$-grid is the $s$-clique extension of the $ttimes t$-grid, if $t$ is large enough. Gavrilyuk and Koo
len used a weaker version of this result to show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection array as a distance-regular graph, if $D$ is large enough.
In this paper, we introduce the concepts of the plain eigenvalue, the main-plain index and the refined spectrum of graphs. We focus on the graphs with two main and two plain eigenvalues and give some characterizations of them.
