On the difference between the spectral radius and the maximum degree of graphs
Let G be a graph with the eigenvalues λ₁(G)≥⋯≥λn(G). The largest eigenvalue of G, λ₁(G), is called the spectral radius of G. Let β(G)=Δ(G)−λ₁(G), where Δ(G) is the maximum degree of vertices of G. It is known that if G is a connected graph, then β(G)≥0 and the equality holds if and only if G is regu...
Збережено в:
| Дата: | 2017 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2017
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/156636 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the difference between the spectral radius and the maximum degree of graphs / M.R. Oboudi // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 302-307. — Бібліогр.: 17 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Резюме: | Let G be a graph with the eigenvalues λ₁(G)≥⋯≥λn(G). The largest eigenvalue of G, λ₁(G), is called the spectral radius of G. Let β(G)=Δ(G)−λ₁(G), where Δ(G) is the maximum degree of vertices of G. It is known that if G is a connected graph, then β(G)≥0 and the equality holds if and only if G is regular. In this paper we study the maximum value and the minimum value of β(G) among all non-regular connected graphs. In particular we show that for every tree T with n≥3 vertices, n−1−√(n−1) ≥ β(T) ≥ 4sin²(π/(2n+2)). Moreover, we prove that in the right side the equality holds if and only if T≅Pn and in the other side the equality holds if and only if T≅Sn, where Pn and Sn are the path and the star on n vertices, respectively. |
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