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Register a new userWhen does a biased graph come from a group labelling?
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A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs on $G$ arise from orienting $G$ and then labelling the edges of $G$ with elements of a group $Gamma$. In this case, we may define a biased graph by declaring a cycle to be balanced if the product of the labels on its edges is the identity, with the convention that we take the inverse value for an edge traversed backwards. Our first result gives a natural topological characterisation of biased graphs arising from group-labellings. In the second part of this article, we use this theorem to construct some exceptional biased graphs. Notably, we prove that for every $m ge 3$ and $ell$ there exists a minor minimal not group labellable biased graph on $m$ vertices where every pair of vertices is joined by at least $ell$ edges. Finally, we show that these results extend to give infinite families of excluded minors for certain families of frame and lift matroids.
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The problem of finding completely positive matrices with equal cp-rank and rank is considered. We give some easy-to-check sufficient conditions on the entries of a doubly nonnegative matrix for it to be completely positive with equal cp-rank and rank. An algorithm is also presented to show its efficiency.
Let $X=(V,E)$ be a finite simple connected graph with $n$ vertices and $m$ edges. A configuration is an assignment of one of two colors, black or white, to each edge of $X.$ A move applied to a configuration is to select a black edge $epsilonin E$ and change the colors of all adjacent edges of $epsilon.$ Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on $X,$ and it corresponds to a group action. This group is called the edge-flipping group $mathbf{W}_E(X)$ of $X.$ This paper shows that if $X$ has at least three vertices, $mathbf{W}_E(X)$ is isomorphic to a semidirect product of $(mathbb{Z}/2mathbb{Z})^k$ and the symmetric group $S_n$ of degree $n,$ where $k=(n-1)(m-n+1)$ if $n$ is odd, $k=(n-2)(m-n+1)$ if $n$ is even, and $mathbb{Z}$ is the additive group of integers.
The [CII] 158$mu$m emission line can arise in all phases of the ISM, therefore being able to disentangle the different contributions is an important yet unresolved problem when undertaking galaxy-wide, integrated [CII] observations. We present a new multi-phase 3D radiative transfer interface that couples Starburst99, a stellar spectrophotometric code, with the photoionisation and astrochemistry codes Mocassin and 3D-PDR. We model entire star forming regions, including the ionised, atomic and molecular phases of the ISM, and apply a Bayesian inference methodology to parametrise how the fraction of the [CII] emission originating from molecular regions, $f_{[CII],mol}$, varies as a function of typical integrated properties of galaxies in the local Universe. The main parameters responsible for the variations of $f_{[CII],mol}$ are specific star formation rate (sSFR), gas phase metallicity, HII region electron number density ($n_e$), and dust mass fraction. For example, $f_{[CII],mol}$ can increase from 60% to 80% when either $n_e$ increases from 10$^{1.5}$ to 10$^{2.5}$cm$^{-3}$, or SSFR decreases from $10^{-9.6}$ to $10^{-10.6}$ yr$^{-1}$. Our model predicts for the Milky Way that $f_{[CII],mol}$$=75.8pm5.9$%, in agreement with the measured value of 75%. When applying the new prescription to a complete sample of galaxies from the Herschel Reference Survey (HRS), we find that anywhere from 60 to 80% of the total integrated [CII] emission arises from molecular regions.
In this paper we show that there does not exist a strongly regular graph with parameters $(1911,270,105,27)$.
A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product G X H of graphs G and H is the graph having vertex set V(G) X V(H) and edge set E(G X H) = {(g_i,h_s)(g_j,h_t): g_ig_j belongs to E(G) and h_sh_t belongs to E(H)}. We prove that the direct product M_m(G) X M_n(H) of the generalized Mycielskians of G and H is a cover graph if and only if G or H is bipartite.
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