Matching Integral Graphs of Small Order

Subject Areas : Statistics

Somayeh Khalashi Ghezelahmad ¹ (Faculty member, Department of Mathematics, Science and Research Branch, Islamic Azad , ,University, Tehran, Iran)

Received: 2019-04-08 Accepted : 2020-01-22 Published : 2020-10-22

Keywords: صحیح تطابقی, چندجمله‌ای تطابقی گراف, تطابق,

Abstract :

In this paper, we study matching integral graphs of small order. A graph is called matching integral if the zeros of its matching polynomial are all integers. Matching integral graphs were first studied by Akbari, Khalashi, etc. They characterized all traceable graphs which are matching integral. They studied matching integral regular graphs. Furthermore, it has been shown that there is no matching integral claw-free graph and K_2 is the only connected matching integral graph with a perfect matching. In this work, we characterize mathching integral graphs according to their order. We determine all connected matching integral graphs on at most seven vertices. We show that there are exactly eight matching integral graphs up to seven vertices. In addition, we prove that any connected matching integral graph on eight vertices has a 3-matching and its size is fourteen. Finally, we characterize all connected matching integral graphs with maximum vertex degrees at most three.

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