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arXiv · 2403.19264

Distinguishing Polynomials of Graphs

Mohammad Hassan Shirdareh Haghighi·Amir Mohammad Ghazanfari·Seyed Ali Reza Talebpour Shirazi Fard
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Abstract

For a graph $G$, a $k$-coloring $c:V(G)\to \{1,2,\ldots, k\}$ is called distinguishing, if the only automorphism $f$ of $G$ with the property $c(v)=c(f(v))$ for every vertex $v\in G$ (color-preserving automorphism), is the identity. In this paper, we show that the number of distinguishing $k$-colorings of $G$ is a monic polynomial in $k$, calling it the distinguishing polynomial of $G$. Furthermore, we compute the distinguishing polynomials of cycles and complete multipartite graphs. We also show that the multiplicity of zero as a root of the distinguishing polynomial of $G$ is at least the number of orbits of $G$.

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Mohammad Hassan Shirdareh Haghighi, Amir Mohammad Ghazanfari, Seyed Ali Reza Talebpour Shirazi Fard. 2024-03-28. Distinguishing Polynomials of Graphs. https://arxiv.org/abs/2403.19264

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