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A note on the Nordhaus-Gaddum type inequality to the second largest eigenvalue of a graph

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Let G be a graph on n vertices and G¯ its complement. In this paper, we prove a Nordhaus-Gaddum type inequality to the second largest eigenvalue of a graph G,… Click to show full abstract

Let G be a graph on n vertices and G¯ its complement. In this paper, we prove a Nordhaus-Gaddum type inequality to the second largest eigenvalue of a graph G, λ2(G), λ2(G) + λ2(G¯) ≤ -1 + √ n2/2-n+1, when G is a graph with girth at least 5. Also, we show that the bound above is tight. Besides, we prove that this result holds for some classes of connected graphs such as trees, k-cyclic, regular bipartite and complete multipartite graphs. Based on these facts, we conjecture that our result holds to any graph.

Keywords: second largest; type inequality; gaddum type; nordhaus gaddum; graph; inequality second

Journal Title: Applicable Analysis and Discrete Mathematics
Year Published: 2017

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