A. Mohammadian

On the Sum of Laplacian Eigenvalues of Graphs

Let k be a natural number and let G be a graph with at least k vertices. A.E. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most e(G) (k choose 2), where e(G) is the number of edges of G. We prove this conjecture for k = 2. We also show that if G is a tree, then the sum of the k largest Laplacian eigenvalues of G is at most e(G) 2k-1.

Commuting Graphs of Matrix Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL n (F) and SL n (F). We show that Γ(M n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL n (F)) and Γ(SL n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M n (F))≃Γ(M m (E)), then n = m and |F|=|E|.

On Commuting Graphs of Finite Matrix Rings

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(M n (F)), for a finite field F and n ≥ 2, then R ≈ M n (F). Here we prove this conjecture when n = 2.

Integral trees of odd diameters

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Recently, Csikvari proved the existence of integral trees of any even diameter. In the odd case, integral trees have been constructed with diameter at most 7. In this article, we show that for every odd integer n>1, there are infinitely many integral trees of diameter n.

Maximum order of trees and bipartite graphs with a given rank

The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order.

Maximum Order of Triangle-Free Graphs with a Given Rank

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle-free graphs with a given rank and characterize all such graphs achieving the maximum order.

On order and rank of graphs

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the number of vertices of every reduced graph of rank r is at most m(r)=2(r+2)/2−2 if r is even and m(r)=5·2(r−3)/2−2 if r is odd. In this article, we prove that if the conjecture is not true, then there would be a counterexample of rank at most 46. We also show that every reduced graph of rank r has at most 8m(r)+14 vertices.

Sums and Products of Square-Zero Matrices

We show that for any two n × n square-zero matrices A and B over a division ring, if the right column spaces of AB and BA are the same, then the rank of AB is at most n/4, and if, in addition, the right null spaces of AB and BA are the same, then the rank of A + B is at most n/2. This generalizes some known results.

Some constructions of integral graphs

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Integral graphs are very rare and difficult to find. In this article, we introduce some general methods for constructing such graphs. As a consequence, some infinite families of integral graphs are obtained.

Hadamard matrices with few distinct types

ABSTRACT The notion of type of quadruples of rows is proven to be useful in the classification of Hadamard matrices. In this paper, we investigate Hadamard matrices with few distinct types. Among other results, the Sylvester Hadamard matrices are shown to be characterized by their spectrum of types.