The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

We prove a conjecture by Van Dam & Sotirov on the smallest eigenvalue of (distance-j) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-j) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical P- and Q-polynomial association schemes

Combinatorial and spectral properties of graphs and association schemes

Cioaba, Sebastian M.The main topics of this dissertation are related to spectral graph theory, a subtopic of algebraic combinatorics. Algebraic combinatorics is the area of mathematics that implements techniques from linear and abstract algebra to solve problems in combinatorics as well as techniques from combinatorics to study algebraic structures. Spectral graph theory focuses on the study of the eigenvalues associated with various matrices for a graph and how the eigenvalues relate to structural properties of the graph. Properties such as connectedness, diameter, independence number, chromatic number and regularity, among others, are all related to the spectrum of a graph. In this dissertation we will study the spectra of various graphs and incorporate well known techniques in spectral graph theory to gain a better understanding of the structure of these graphs. We focus on three topics (Chapters 2, 3 and 4). The variation of these topics reinforces how diverse and useful spectral techniques in graph theory can be. ☐ In Chapter 1 we cover notation and basic definitions used throughout this dissertation. We also introduce some well known and powerful results relating the structural properties of a graph to its spectrum. This includes a discussion of the properties of graphs that can be established from the spectrum as well as which graphs are determined by their spectrum. Finally, we give definitions and basic results for association schemes and distance-regular graphs since the results of this dissertation are related to these structures. ☐ In Chapter 2 we study the smallest eigenvalues for distance-j graphs in both the Hamming and Johnson association schemes. Our results for distance-j Hamming graphs settle a conjecture proposed by Van Dam and Sotirov in [29]. In fact, we reach a stronger conclusion than the one proposed by Van Dam and Sotirov. Our results for x distance-j Johnson graphs settle a conjecture proposed by Karloff in [46]. Again, we are able to obtain a stronger conslusion than what is presented in Karloff's conjecture. ☐ In Chapter 3 we use the technique of Godsil-McKay switching to construct cospectral mates for graphs formed by taking the union of relations in the Johnson association scheme. Our results offer insight into which graphs in this scheme are not determined by their spectrum. Our work also unifies the switching sets previously found for Johnson graphs in [26] and Kneser graphs in [43]. We also present some open problems related to our work, including a switching set that we would like to see generalized in order to obtain a new infinite family of graphs in the Johnson scheme that are not determined by their spectrum. ☐ In Chapter 4 we examine connectivity properties of distance-regular graphs and graphs related to association schemes. In particular, we prove a result on the minimum number of edges that need to be deleted from a distance-regular graph in order to disconnect it into nonsingleton components. We also prove a result on the edge-connectivity of distance-j twisted Grassmann graphs which supports a conjecture proposed by Godsil in [35]. Finally, we end the chapter by presenting open problems dealing with the connectivity of color classes in association schemes.University of Delaware, Department of Mathematical SciencesPh.D

The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

\u3cp\u3eWe prove a conjecture by Van Dam & Sotirov on the smallest eigenvalue of (distance-j) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-j) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical P- and Q-polynomial association schemes.\u3c/p\u3

Cospectral mates for the union of some classes in the Johnson association scheme

Let $n\geq k\geq 2$ be two integers and $S$ a subset of $\{0,1,\dots,k-1\}$. The graph $J_{S}(n,k)$ has as vertices the $k$-subsets of the $n$-set $[n]=\{1,\dots,n\}$ and two $k$-subsets $A$ and $B$ are adjacent if $|A\cap B|\in S$. In this paper, we use Godsil-McKay switching to prove that for $m\geq 0$, $k\geq \max(m+2,3)$ and $S = \{0, 1, ..., m\}$, the graphs $J_S(3k-2m-1,k)$ are not determined by spectrum and for $m\geq 2$, $n\geq 4m+2$ and $S = \{0,1,...,m\}$ the graphs $J_{S}(n,2m+1)$ are not determined by spectrum. We also report some computational searches for Godsil-McKay switching sets in the union of classes in the Johnson scheme for $k\leq 5$.Comment: 9 pages, no figures, 3 tables; 2nd version contains improved results compared to the 1st versio

Cospectral mates for the union of some classes in the Johnson association scheme

Let n≥k≥2 be two integers and S a subset of {0,1,…,k−1}. The graph JS(n,k) has as vertices the k-subsets of the n-set [n]={1,…,n} and two k-subsets A and B are adjacent if |A∩B|∈S. In this paper, we use Godsil–McKay switching to prove that for m≥0, k≥max⁡(m+2,3) and S={0,1,…,m}, the graphs JS(3k−2m−1,k) are not determined by spectrum and for m≥2, n≥4m+2 and S={0,1,…,m} the graphs JS(n,2m+1) are not determined by spectrum. We also report some computational searches for Godsil–McKay switching sets in the union of classes in the Johnson scheme for k≤5