The graphs with all but two eigenvalues equal to $$$$ or 0

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to \(-2\), or 0, and determine which of these graphs are determined by their adjacency spectrum.     

Cospectral graphs and the generalized adjacency matrix

Let J be the all-ones matrix, and let A denote the adjacency matrix of a graph. An old result of Johnson and Newman states that if two graphs are cospectral with respect to yJ − A for two distinct values of y, then they are cospectral for all y. Here we will focus on graphs cospectral with respect to yJ − A for exactly one value of y. We call such graphs -cospectral. It follows that is a rational number, and we prove existence of a pair of -cospectral graphs for every rational . In addition, we generate by computer all -cospectral pairs on at most nine vertices. Recently, Chesnokov and the second author constructed pairs of -cospectral graphs for all rational , where one graph is regular and the other one is not. This phenomenon is only possible for the mentioned values of , and by computer we find all such pairs of -cospectral graphs on at most eleven vertices.     

Universal adjacency matrices with two eigenvalues

Consider a graph Γ on n vertices with adjacency matrix A and degree sequence (d₁,…,d_n). A universal adjacency matrix of Γ is any matrix in Span {A,D,I,J} with a nonzero coefficient for A, where and I and J are the n×n identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur.     

The maximum order of adjacency matrices of graphs with a given rank

We look for the maximum order m(r) of the adjacency matrix A of a graph G with a fixed rank r, provided A has no repeated rows or all-zero row. Akbari, Cameron and Khosrovshahi conjecture that m(r)?=?2^(r+2)/2 ? 2 if r is even, and m(r)?=?5 · 2^(r?3)/2 ? 2 if r is odd. We prove the conjecture and characterize G in the case that G contains an induced subgraph ${\frac{r}{2}K_2}$ or ${\frac{r-3}{2}K_2+K_3}$ .     

The chemical properties of 1-substituted 4-oxopyrimidinium cations

Some theoretical aspects of electrophilic properties of 4-pyrimidinium cations are discussed. The reactivity towards a series of C-, H-, N- and O-nucleophiles is studied.     

Polymer-assisted solution-phase parallel synthesis of dipeptide p-nitroanilides and dipeptide diphenyl phosphonates

This letter describes the parallel synthesis of dipeptide p-nitroanilides (1) and dipeptide diphenyl phosphonates (2), compounds that can be used as substrates and irreversible inhibitors for the rapid profiling of dipeptidyl peptidases. A polymer-assisted solution-phase synthesis was used for a rapid and clean coupling between easily available building blocks.     

A revision of the alcoholysis and alkanethiolysis of 3‐amino‐6‐chloropyridazine

The synthesis of 3‐amino‐6‐alkoxy‐ and 3‐amino‐6‐alkylthiopyridazines via nucleophilic aromatic substitution on 3‐amino‐6‐chloropyridazine is described. In contrast to literature reports, no pressure tube is required to perform these reactions.     

A Characterization of Distance-Regular Graphs with Diameter Three

We characterize the distance-regular graphs with diameter three by giving an expression for the number of vertices at distance two from each given vertex, in terms of the spectrum of the graph.     

The Search for Pseudo Orthogonal Latin Squares of Order Six

We report on the completecomputer search for a strongly regular graph with parameters(36,15,6,6) and chromatic number six. The resultis that no such graph exists.