Eigenvalues and triangles in graphs

On Tensor Product of Graphs, Girth and Triangles

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph $G 1 K_2$ to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.

Triangles in random graphs

Triangles in Random Graphs

OF THE DISSERTATION Triangles in Random Graphs by Robert DeMarco Dissertation Director: Jeff Kahn We prove four separate results. These results will appear or have appeared in various papers (see [10], [11], [12], [13]). For a gentler introduction to these results, the reader is directed to the first chapter of this thesis. Let G = Gn,p. With ξk = ξ n,p k the number of copies of Kk in G, p ≥ n−...

On the distance eigenvalues of Cayley graphs

In this paper, we determine the distance matrix and its characteristic polynomial of a Cayley graph over a group G in terms of irreducible representations of G. We give exact formulas for n-prisms, hexagonal torus network and cubic Cayley graphs over abelian groups. We construct an innite family of distance integral Cayley graphs. Also we prove that a nite abelian group G admits a connected...

Dirichlet eigenvalues of asymptotically flat triangles

This paper is devoted to the study of the eigenpairs of the Dirichlet Laplacian on a family of triangles where two vertices are fixed and the altitude associated with the third vertex goes to zero. We investigate the dependence of the eigenvalues on this altitude. For the first eigenvalues and eigenfunctions, we obtain an asymptotic expansion at any order at the scale cube root of this altitude...