Set Representations of Linegraphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A family $\mathcal{S}$ of nonempty sets $\{S_1,\ldots,S_n\}$ is a set representation of $G$ if there exists a one-to-one correspondence between the vertices $v_1, \ldots, v_n$ in $V(G)$ and the sets in $\mathcal{S}$ such that $v_iv_j \in E(G)$ if and only if S_i\cap S_j\neq \es. A set representation $\mathcal{S}$ is a distinct (respectively, antichain, uniform and simple) set representation if any two sets $S_i$ and $S_j$ in $\mathcal{S}$ have the property $S_i\neq S_j$ (respectively, $S_i\nsubseteq S_j$, $|S_i|=|S_j|$ and $|S_i\cap S_j|\leqslant 1$). Let $U(\mathcal{S})=\bigcup_{i=1}^n S_i$. Two set representations $\mathcal{S}$ and $\mathcal{S}'$ are isomorphic if $\mathcal{S}'$ can be obtained from $\mathcal{S}$ by a bijection from $U(\mathcal{S})$ to $U(\mathcal{S}')$. Let $F$ denote a class of set representations of a graph $G$. The type of $F$ is the number of equivalence classes under the isomorphism relation. In this paper, we investigate types of set representations for linegraphs. We determine the types for the following categories of set representations: simple-distinct, simple-antichain, simple-uniform and simple-distinct-uniform

Constraints on Axion-like Particles from Observations of Mrk 421 using the CLs{\rm CL_s} Method

Axion-like particles (ALPs) could mix with photons in the presence of astrophysical magnetic fields, and result in oscillations in the high energy $\gamma$-ray spectra observed by experiments. In this work, we investigate the ALP-photon oscillation effect through the blazar Mrk 421 spectra of 15 periods observed by Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC) and Fermi Large Area Telescope (Fermi-LAT). Compared with previous studies, we generate the mock data under the ALP hypothesis and apply the ${\rm CL_s}$ method to set constraints on the ALP parameters. This method is widely employed in high energy experiments and could avoid the possibility of excluding some parameter regions due to the fluctuation. We find that the ALP-photon coupling $g_{a\gamma}$ is constrained to be smaller than $\sim 2\times10^{-11}$ GeV$^{-1}$ for ALP mass ranging from $10^{-9}$ eV to $10^{-7}$ eV at a 95\% confidence level. The constraints obtained with the method based on the TS distribution under the null hypothesis, which is adopted in many previous astrophysical ALP studies, are also shown. Our results demonstrate that the joint constraints of all the periods from both methods are consistent. However, the latter method fails to provide constraints for some observation periods, whereas the ${\rm CL_s}$ method remains effective in such cases.Comment: 10 pages, 26 figure