The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and the average degree of the vertices of G. Motivated by the work from Sharafdini an...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...

Let G be a graph of su¢ ciently large order n; and let the largest eigenvalue (G) of its adjacency matrix satis…es (G) > p bn2=4c: Then G contains a cycle of length t for every t n=320: This condition is sharp: the complete bipartite graph T2 (n) with parts of size bn=2c and dn=2e contains no odd cycles and its largest eigenvalue is equal to p bn2=4c: This condition is stable: if (G) is close t...

The k-diameter of a graph 0 is the largest pairwise minimum distance of a set of k vertices in 0, i.e., the best possible distance of a code of size k in 0. A k-diameter for some k is called a multidiameter of the graph. We study the function N (k,1, D), the largest size of a graph of degree at most1 and kdiameter D. The graphical analogues of the Gilbert bound and the Hamming bound in coding t...

We developed a procedure of reducing the number of vertices and edges of a given tree, which we call the " tree simplification procedure, " without changing its topological information. Our motivation for developing this procedure was to reduce computational costs of graph Laplacian eigenvalues of such trees. When we applied this procedure to a set of trees representing dendritic structures of ...

You may have figured out by now that I like cutting graphs into pieces. It is how I discover the structure of a graph. When I encounter a new graph and want to understand what it looks like I first try drawing it. If the drawing looks good, I feel like I understand it. If the drawing looks good, I try chopping the graph into pieces without cutting too many edges. That is, I look for cuts of low...

Let G be a graph of order n and let μ be an eigenvalue of multiplicity m. A star complement for μ in G is an induced subgraph of G of order n−m with no eigenvalue μ. In this paper, we study the maximal graphs as well as regular graphs which have Kr,s + tK1 as a star complement for eigenvalue 1. It turns out that some well known strongly regular graphs are uniquely determined by such a star comp...

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise...

Let r and m be two integers such that r > m. Let H be a graph with order |H|, size e and maximum degree r such that 2e > |H|r−m. We find a best lower bound on spectral radius of graph H in terms of m and r. Let G be a connected r-regular graph of order |G| and k < r be an integer. Using the previous results, we find some best upper bounds (in terms of r and k) on the third largest eigenvalue th...

Our research focuses on developing design-oriented analytical tools that enable us to better understand how a network comprising dynamic and static elements behaves when it is set in oscillatory motion, and how the interconnection topology relates to the spectral properties of the system. Such oscillatory networks are ubiquitous, extending from miniature electronic circuits to large-scale power...