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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)
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 Pd−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = [p1, p2, ..., pd−1] such that p1 ≥ 1, p2 ≥ 1, ..., pd−1 ≥ 1. Let C (p) be the caterpillar obtained from the stars Sp1 , Sp2 , ..., Spd−1 and the path Pd−1 by(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)