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Paper IPM / M / 15372  


Abstract:  
A starlike tree is a tree with exactly one vertex of degree greater than two. The spectral radius of a graph G, that is denoted by λ(G), is the largest eigenvalue of G.
Let k and n_{1},…,n_{k} be some positive integers. Let T(n_{1},…,n_{k}) be the tree T ( T is a path or a starlike tree) such that T has a vertex v so that T\v is the disjoint union of the paths P_{n1−1},…,P_{nk−1} where every neighbor of v in T has degree one or two.
Let P=(p_{1},…,p_{k}) and
Q=(q_{1},…,q_{k}), where p_{1} ≥ … ≥ p_{k} ≥ 1 and
q_{1} ≥ … ≥ q_{k} ≥ 1 are integer. We say P majorizes Q and let P\succeq_{M} Q,
if for every j, 1 ≤ j ≤ k,
∑_{i=1}^{j}p_{i} ≥ ∑_{i=1}^{j}q_{i}, with equality if j=k.
In this paper we show that if P majorizes Q, that is (p_{1},…,p_{k})\succeq_{M}(q_{1},…,q_{k}), then λ(T(q_{1},…,q_{k})) ≥ λ(T(p_{1},…,p_{k})).
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