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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)
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)
A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Gm be a connected weighted graph on m vertices. Let (iii) the edges of B i joining the vertices at the level j with the vertices at the level (j + 1) have weight w i,k i −j for j = 1, 2,. .. , k i − 1. Let Gm {B i : 1 ≤ i ≤ m} be the(More)