We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This affirmatively answers a question of Chung and Graham (2002) for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.
Let be a tree with vertices. To each edge of we assign a weight which is a positive definite matrix of some fixed order, say, . Let denote the sum of all the weights lying in the path connecting the vertices and of . We now say that is the distance between and . Define , where is the null matrix and for , is the distance between and . Let be an arbitrary connected weighted graph with vertices, where each weight is a positive definite matrix of order . If and...
Let and be points in . Write if is a multiple of . Two different points and in uniquely determine a tropical line passing through them and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on leaves. It is also a metric graph. If some representatives and of and are the first and second columns of some real normal idempotent order matrix , we prove that the tree is described by a matrix , easily obtained from . We also prove that...
For a given complex square matrix with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first, we derive bounds for the second largest and the smallest eigenvalues of adjacency matrices of -regular graphs. Then we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest and the largest eigenvalues of the Laplacian matrices of graphs. The sharpness of these bounds is verified...
Necessary conditions for an undirected graph G to contain a graph H as induced subgraph involving the smallest ordinary or the largest normalized Laplacian eigenvalue of G are presented.
In this paper equienergetic self-complementary graphs on vertices for every , and , are constructed.
We consider the primitive two-colored digraphs whose uncolored digraph has vertices and consists of one -cycle and one -cycle. We give bounds on the exponents and characterizations of extremal two-colored digraphs.