2021
DOI: 10.3390/sym13040710
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Orbit Polynomial of Graphs versus Polynomial with Integer Coefficients

Abstract: Suppose ai indicates the number of orbits of size i in graph G. A new counting polynomial, namely an orbit polynomial, is defined as OG(x) = ∑i aixi. Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials.

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Cited by 4 publications
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“…The automorphism group of this graph is S 1 , which implies that O F (x) = 115x and O F (x) = 1 − 115x, which yield that δ = 0.008. In Table 1, some results including the sizes of the networks, the orders of the automorphism groups, and the positive roots of the modified orbit polynomials [22] are summarized. In this table, the symbol δ denotes the unique positive root of the modified orbit polynomial.…”
Section: Application In Real-world Networkmentioning
confidence: 99%
“…The automorphism group of this graph is S 1 , which implies that O F (x) = 115x and O F (x) = 1 − 115x, which yield that δ = 0.008. In Table 1, some results including the sizes of the networks, the orders of the automorphism groups, and the positive roots of the modified orbit polynomials [22] are summarized. In this table, the symbol δ denotes the unique positive root of the modified orbit polynomial.…”
Section: Application In Real-world Networkmentioning
confidence: 99%