Bounding the largest eigenvalue of trees in terms of the largest vertex degree

Let <f>λ1(G)</f> denote the largest eigenvalue of the adjacency matrix and let <f>μ1(G)</f> denote the largest eigenvalue of the Laplacian matrix of a graph G. It is well known that if a graph G has the largest vertex degree <f>Δ≠0</f> then√ of <RCD>Δ</RCD></rad>&les;λ1(G)&les;Δ<hsp sp="0.35"><rm>and</rm><hsp sp="0.35">Δ+1&les;μ1(G)&les;2Δ.</fd>Thus the gap between the maximum and minimum value of <f>λ1(G)</f> and <f>μ1(G)</f> in the class of graphs with fixed <f>Δ</f> is <f>Θ(Δ)</f>. In this note we show that in the class of trees with fixed <f>Δ</f> this gap is just <f>Θ(<rad><RCD>Δ</RCD></rad>)</f>. Namely, we show that if a tree <it>T</it> has the largest vertex degree <f>Δ</f> then<fd>λ1(T)<2<rad><RCD>Δ−1</RCD></rad><hsp sp="0.35"><rm>and</rm><hsp sp="0.35">μ1(T)<Δ+2<rad><RCD>Δ−1</RCD>&les;λ1(G)&les;Δ and Δ+1&les;μ1(G)&les;2Δ.Thus the gap between the maximum and minimum value of <f>λ1(G)</f> and <f>μ1(G)</f> in the class of graphs with fixed <f>Δ</f> is <f>Θ(Δ)</f>. In this note we show that in the class of trees with fixed <f>Δ</f> this gap is just <f>Θ(√ of <RCD>Δ</RCD></rad>)</f>. Namely, we show that if a tree <it>T</it> has the largest vertex degree <f>Δ</f> then<fd>λ1(T)<2<rad><RCD>Δ−1</RCD></rad><hsp sp="0.35"><rm>and</rm><hsp sp="0.35">μ1(T)<Δ+2<rad><RCD>Δ−1</RCD>)</f>. Namely, we show that if a tree T has the largest vertex degree <f>Δ</f> thenλ1(T)<2√ of <RCD>Δ−1</RCD></rad><hsp sp="0.35"><rm>and</rm><hsp sp="0.35">μ1(T)<Δ+2<rad><RCD>Δ−1</RCD> and μ1(T)<Δ+2√ of <RCD>Δ−1</RCD>.New bounds are an improvement for <f>Δ&ges;3</f>. [Copyright &y& Elsevier]