Let G be a simple, undirected, connected and finite graph. with the pendent vertices of K 1,si. It is proved that (i) if µ ̸ = 0 and µ ̸ = 1 is a Laplacian eigenvalue of G, then µ is a Laplacian eigenvalue of G(H 1 ,. .. , H r) and (ii) for 1 ≤ i ≤ r, if µ is a Laplacian eigenvalue of H i , µ ̸ = 0 or µ = 0 with an eigenvector orthogonal to the all ones… (More)
A caterpillar is a tree in which the removal of all pendent vertices make it a path. In this paper, we consider two classes of caterpillars. We present an ordering of caterpillars by algebraic connectivity in one of them and find one that maximizes the algebraic connectivity in the other class.
Bose-Einstein condensation (BEC) in two dimensions (2D) (e.g., to describe the quasi-2D cuprates) is suggested as the possible mechanism widely believed to underlie superconduc-tivity in general. A crucial role is played by nonzero center-of-mass momentum Cooper pairs (CPs) usually neglected in BCS theory. Also vital is the unique linear dispersion relation… (More)
Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α ∈ [0, 1], write A α (G) for the matrix A α (G) = αD (G) + (1 − α)A (G). Let α 0 (G) be the smallest α for which A α (G) is positive semidefinite. It is known that α 0 (G) ≤ 1/2. The main results of this paper are: (1) if G is d-regular then… (More)
Let G be a simple undirected connected graph on n vertices. Suppose that the vertices of G are labelled 1,2,... ,n. Let d i be the degree of the vertex i. The Randi´c matrix of G , denoted by R, is the n × n matrix whose (i, j) − entry is 1 √ d i d j if the vertices i and j are adjacent and 0 otherwise. The normalized Laplacian matrix of G is L = I − R,… (More)
A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let P d−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. In this paper, the caterpillars in C and in S having the maximum and the minimum algebraic connectivity are found. Moreover, the algebraic connectivity of a… (More)