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Polycyclic Aromatic Hydrocarbons (PAHs) are ubiquitous complex molecules in the interstellar medium and are used as an indirect indicator of star-formation. On the other hand the ultraviolet (UV) emission from the young massive stars directly traces the star formation activity in a galaxy. The James Webb Space Telescope (JWST), along with the UltraViolet Imaging Telescope (UVIT), opened up a new window of opportunity to make a better understanding of the properties of the PAH molecules associated with the star-forming regions. In this study, we investigate how the resolved scale properties of PAH molecules in nearby galaxies are affected by star-formation. We analyze the PAH features observed at 3.3, 7.7, and 11.3 {\mu}m using F335M, F770W, and F1130W images obtained from JWST. Additionally, we utilize UVIT images to assess the star formation associated with these PAH emitting regions. Our study focuses on three galaxies, namely NGC 628, NGC 1365, and NGC 7496, selected based on the availability of both JWST and UVIT images. Based on the resolved scale study on the PAH bright regions using JWST and UVIT images, we found that the fraction of ionized PAH molecules is high in the star-forming regions with high {\Sigma}SFR. We observed that emission from smaller PAH molecules is more in the star-forming regions with higher {\Sigma}SFR. Our study suggests that the PAH molecules excited by the photons from SF regions with higher {\Sigma}SFR are dominantly smaller and ionized molecules. UV photons from the star-forming regions could be the reason for a higher fraction of the ionized PAHs. We suggest that the effect of high temperature in the star-forming regions and the formation of smaller PAH molecules in the star-forming regions might also be resulting in the higher fraction of emission in the F335MPAH band., Comment: Accepted for publication in Astronomy & Astrophysics

During the visual observations of optical imaging data obtained from the DECaLS, a serendipitous discovery emerged, revealing the presence of a ringed galaxy, DES J024008.08-551047.5 (DJ0240). We performed one dimensional isophotal and two-dimensional GALFIT analysis to confirm the orthogonal nature of the ring galaxy and identify distinct components within the host galaxy. We discovered the galaxy DJ0240 as a potential PRG candidate with a ring component positioned almost perpendicular to the host galaxy. The position angles of the ring and host components have been determined to be 80 and 10 degrees, respectively, indicating that they are nearly orthogonal to each other. We observed that the ring component extends three times more than the host galaxy and shows a distinct colour separation, being bluer than the host. The estimated g - r colour values of host and ring components are 0.86+/-0.02 and 0.59+/-0.10 mag, respectively. The colour value of the ring component is similar to typical spiral galaxies. The host galaxy`s colour and the presence of a bulge and disk components indicate the possibility of the host galaxy being a lenticular type. Based on the comparison of photometric properties between the PRGs and other ring-type galaxies (RTGs), our findings reveal a subtle, yet noticeable, colour difference between the host and ring components. We observed that both host and ring components of DJ0240 align more closely with PRGs than with RTGs. Furthermore, we compared the sersic index values of the ring component (nring) of galaxy DJ0240 with a selected sample of PRGs and Hoag-type galaxies. The results showed DJ0240 had a remarkably low nring value of 0.13, supporting the galaxy`s classification as a PRG. Hence, we suggest that the ring galaxy DJ0240 is a highly promising candidate for inclusion in the family of PRGs., Comment: 5 pages, 5 figures, accepted for publication in A&A journal

Motivated by the famous Hadwiger's Conjecture, we study the properties of $8$-contraction-critical graphs with no $K_7$ minor; we prove that every $8$-contraction-critical graph with no $K_7$ minor has at most one vertex of degree $8$, where a graph $G$ is $8$-contraction-critical if $G$ is not $7$-colorable but every proper minor of $G$ is $7$-colorable. This is one step in our effort to prove that every graph with no $K_7$ minor is $7$-colorable, which remains open.

Secular and environmental effects play a significant role in regulating the star formation rate and hence the evolution of the galaxies. Since UV flux is a direct tracer of the star formation in galaxies, the UltraViolet Imaging Telescope (UVIT) onboard ASTROSAT enables us to characterize the star forming regions in a galaxy with its remarkable spatial resolution. In this study, we focus on the secular evolution of NGC 628, a spiral galaxy in the local universe. We exploit the resolution of UVIT to resolve up to $\sim$ 63 pc in NGC 628 for identification and characterization of the star forming regions. We identify 300 star forming regions in the UVIT FUV image of NGC 628 using ProFound and the identified regions are characterized using Starburst99 models. The age and mass distribution of the star forming regions across the galaxy supports the inside-out growth of the disk. We find that there is no significant difference in the star formation properties between the two arms of NGC 628. We also quantify the azimuthal offset of the star forming regions of different ages. Since we do not find an age gradient, we suggest that the spiral density waves might not be the possible formation scenario of the spiral arms of NGC 628. The headlight cloud present in the disk of the galaxy is found to be having the highest star formation rate density ($0.23 M_{\odot} yr^{-1} kpc^{-2}$) compared to other star forming regions on spiral arms and the rest of the galaxy., Comment: 11 pages, 11 Figures, Accepted in MNRAS

Let $G$ be a plane graph with $C$ the boundary of the outer face and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $\phi$ of $J$ such that $\phi(v)\in L(v)$ for every vertex $v$ of $J$. Thomassen proved that if $v_1,v_2\in V(C)$ are adjacent, $L(v_1)\ne L(v_2)$, $|L(v)|\ge3$ for every $v\in V(C)\setminus \{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\in V(G)\setminus V(C)$, then $G$ has an $L$-coloring. As one final application in this last part of our series on $5$-list-coloring, we derive from all of our theory a far-reaching generalization of Thomassen's theorem, namely the generalization of Thomassen's theorem to arbitrarily many such faces provided that the faces are pairwise distance $D$ apart for some universal constant $D>0$., Comment: 14 pages

A cornerstone theorem in the Graph Minors series of Robertson and Seymour is the result that every graph $G$ with no minor isomorphic to a fixed graph $H$ has a certain structure. The structure can then be exploited to deduce far-reaching consequences. The exact statement requires some explanation, but roughly it says that there exist integers $k,n$ depending on $H$ only such that $0

It is known that $A$-paths of length $0$ mod $m$ satisfy the Erd\H{o}s-P\'osa property if $m=2$ or $m=4$, but not if $m > 4$ is composite. We show that if $p$ is prime, then $A$-paths of length $0$ mod $p$ satisfy the Erd\H{o}s-P\'osa property. More generally, in the framework of undirected group-labelled graphs, we characterize the abelian groups $\Gamma$ and elements $\ell \in \Gamma$ for which the Erd\H{o}s-P\'osa property holds for $A$-paths of weight $\ell$.

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs $(G,\gamma)$ where $\gamma$ assigns to each edge of an undirected graph $G$ an element of an abelian group $\Gamma$. As a consequence, we prove that $\Gamma$-nonzero cycles (cycles whose edges sum to a non-identity element of $\Gamma$) satisfy the half-integral Erd\H{o}s-P\'osa property, and we also recover a result of Wollan that, if $\Gamma$ has no element of order two, then $\Gamma$-nonzero cycles satisfy the Erd\H{o}s-P\'osa property. As another application, we prove that if $m$ is an odd prime power, then cycles of length $\ell \mod m$ satisfy the Erd\H{o}s-P\'osa property for all integers $\ell$. This partially answers a question of Dejter and Neumann-Lara from 1987 on characterizing all such integer pairs $(\ell,m)$.

Robertson and Seymour's celebrated Graph Minor Theorem states that graphs are well-quasi-ordered by the minor relation. Unlike the minor relation, the topological minor relation does not well-quasi-order graphs in general. Among all known infinite antichains with respect to the topological containment, subdivisions of a graph obtained from an arbitrarily long path by duplicating each edge can be found. In the 1980's Robertson conjectured that this is the only obstruction. Formally, he conjectured that for every positive integer $k$, graphs that do not contain the graph obtained from a path of length $k$ by duplicating each edge as a topological minor are well-quasi-ordered by the topological minor relation. The case $k=1$ implies Kruskal's Tree Theorem, and the case $k=2$ implies a conjecture of V\'{a}zsonyi on subcubic graphs. This series of papers dedicates a proof of Robertson's conjecture. We prove Robertson's conjecture for graphs of bounded tree-width in this paper. It is an essential step toward the complete proof of Robertson's conjecture, and the machinery developed in this paper will be applied in future papers of the series. This bounded tree-width case proved in this paper implies all known results about well-quasi-ordering graphs by the topological minor relation that can be proved without using the Graph Minor Theorem, and our proof in this paper is self-contained.

A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick is minimal if for every edge e the deletion of e results in a graph that is not a brick. We prove a generation theorem for minimal bricks and two corollaries: (1) for n>4, every minimal brick on 2n vertices has at most 5n-7 edges, and (2) every minimal brick has at least three vertices of degree three., Comment: 10 pages, 2 figures. This version fixes an error kindly pointed to us by P. A. Fabres, N. Kothari and M. H. de Carvalho

Depression is one of the most common and a major concern for society. Proper monitoring using devices that can aid in its detection could be helpful to prevent it all together. The Distress Analysis Interview Corpus (DAIC) is used to build a metric-based depression detection. We have designed a metric to describe the level of depression using negative sentences and classify the participant accordingly. The score generated from the algorithm is then levelled up to denote the intensity of depression. The results show that measuring depression is very complex to using text alone as other factors are not taken into consideration. Further, In the paper, the limitations of measuring depression using text are described, and future suggestions are made., Comment: 7 pages, 1 Table

We prove two results: 1. A graph $G$ on at least seven vertices with a vertex $v$ such that $G-v$ is planar and $t$ triangles satisfies $|E(G)| \leq 3|V(G)|- 9 + t/3$. 2. For $p=2,3,\ldots,9$, a triangle-free graph $G$ on at least $2p-5$ vertices with no $K_p$-minor satisfies $|E(G)|\leq (p-2)|V(G)| - (p-2)^2$., Comment: 12 pages, no figures. Revised according to referees' comments

Let $P$ be a graph with a vertex $v$ such that $P\backslash v$ is a forest, and let $Q$ be an outerplanar graph. We prove that there exists a number $p=p(P,Q)$ such that every 2-connected graph of path-width at least $p$ has a minor isomorphic to $P$ or $Q$. This result answers a question of Seymour and implies a conjecture of Marshall and Wood. The proof is based on a new property of tree-decompositions., Comment: 34 pages, 5 figures, this revision includes the contents of arXiv:1712.00653

Let $G=(V,E)$ be a finite undirected graph. If $P$ is an oriented path from $r_1\in V$ to $r_2\in V$, we define $\partial(P) = r_2-r_1$. If $R, S\subseteq V$, we denote by $P(G; R, S)$ the span of the set of all $\partial P\otimes \partial Q$ with $P$ and $Q$ disjoint oriented paths of $G$ connecting vertices in $R$ and $S$, respectively. By $L(R, S)$, we denote the submodule of $\mathbb{Z}\langle R\rangle\otimes\mathbb{Z}\langle S\rangle$ consisting all $\sum_{r\in R, s\in S} c(r,s)r\otimes s$ such that $c(r,r) = 0$ for all $r\in R\cap S$, $\sum_{r\in R} c(r, s) = 0$ for all $s\in S$, and $\sum_{s\in S} c(r, s) = 0$ for all $r\in R$. In this paper, we provide, when $G$ is sufficiently connected, characterizations when $P(G; R, S)$ is a proper subset of $L(R, S)$.

We prove that if a graph has a tree-decomposition of width at most w, then it has a tree-decomposition of width at most w with certain desirable properties. We will use this result in a subsequent paper to show that every 2-connected graph of large path-width has a minor isomorphic to either a large tree with a vertex attached to every vertex of the tree or a large outerplanar graph., Comment: The paper has been subsumed into arXiv:1712.04549

Let $G=(V,E)$ be a finite undirected graph. Orient the edges of $G$ in an arbitrary way. A $2$-cycle on $G$ is a function $d : E^2\to \mathbb{Z}$ such for each edge $e$, $d(e, \cdot)$ and $d(\cdot, e)$ are circulations on $G$, and $d(e, f) = 0$ whenever $e$ and $f$ have a common vertex. We show that each $2$-cycle is a sum of three special types of $2$-cycles: cycle-pair $2$-cycles, Kuratowski $2$-cycles, and quad $2$-cycles. In case that the graph is Kuratowski connected, we show that each $2$-cycle is a sum of cycle-pair $2$-cycles and at most one Kuratowski $2$-cycle. Furthermore, if $G$ is Kuratowski connected, we characterize when every Kuratowski $2$-cycle is a sum of cycle-pair $2$-cycles. A $2$-cycles $d$ on $G$ is skew-symmetric if $d(e,f) = -d(f,e)$ for all edges $e,f\in E$. We show that each $2$-cycle is a sum of two special types of skew-symmetric $2$-cycles: skew-symmetric cycle-pair $2$-cycles and skew-symmetric quad $2$-cycles. In case that the graph is Kuratowski connected, we show that each skew-symmetric $2$-cycle is a sum of skew-symmetric cycle-pair $2$-cycles. Similar results like this had previously been obtained by one of the authors for symmetric $2$-cycles. Symmetric $2$-cycles are $2$-cycles $d$ such that $d(e,f)=d(f,e)$ for all edges $e,f\in E$.

An embedding of a graph in $3$-space is linkless if for every two disjoint cycles there exists an embedded ball that contains one of the cycles and is disjoint from the other. We prove that every bipartite linklessly embeddable (simple) graph on $n\ge5$ vertices has at most $3n-10$ edges, unless it is isomorphic to the complete bipartite graph $K_{3,n-3}$., Comment: 20 pages; revised according to referees' comments

We prove an ear-decomposition theorem for $4$-edge-connected graphs and use it to prove that for every $4$-edge-connected graph $G$ and every $r\in V(G)$, there is a set of four spanning trees of $G$ with the following property. For every vertex in $G$, the unique paths back to $r$ in each tree are edge-disjoint. Our proof implies a polynomial-time algorithm for constructing the trees., Comment: 22 pages, 4 figures. Presented at the 29th Cumberland Conference on Combinatorics, Graph Theory and Computing at Vanderbilt University

Let $G$ be a graph embedded in a fixed surface $\Sigma$ of genus $g$ and let $L=(L(v):v\in V(G))$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $G$ is triangle-free, or each list has size at least three and $G$ has no cycle of length four or less. An $L$-coloring of $G$ is a mapping $\phi$ with domain $V(G)$ such that $\phi(v)\in L(v)$ for every $v\in V(G)$ and $\phi(v)\ne\phi(u)$ for every pair of adjacent vertices $u,v\in V(G)$. We prove * if every non-null-homotopic cycle in $G$ has length $\Omega(\log g)$, then $G$ has an $L$-coloring, * if $G$ does not have an $L$-coloring, but every proper subgraph does ("$L$-critical graph"), then $|V(G)|=O(g)$, * if every non-null-homotopic cycle in $G$ has length $\Omega(g)$, and a set $X\subseteq V(G)$ of vertices that are pairwise at distance $\Omega(1)$ is precolored from the corresponding lists, then the precoloring extends to an $L$-coloring of $G$, * if every non-null-homotopic cycle in $G$ has length $\Omega(g)$, and the graph $G$ is allowed to have crossings, but every two crossings are at distance $\Omega(1)$, then $G$ has an $L$-coloring, and * if $G$ has at least one $L$-coloring, then it has at least $2^{\Omega(|V(G)|)}$ distinct $L$-colorings. We show that the above assertions are consequences of certain isoperimetric inequalities satisfied by $L$-critical graphs, and we study the structure of families of embedded graphs that satisfy those inequalities. It follows that the above assertions hold for other coloring problems, as long as the corresponding critical graphs satisfy the same inequalities., Comment: 65 pages, revised based on referees' comments

Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of $G$ we mean a (proper) coloring $\phi$ of $G$ such that $\phi(v)\in L(v)$ for every vertex $v$ of $G$. Thomassen proved that if $v_1,v_2\in V(C)$ are adjacent, $L(v_1)\ne L(v_2)$, $|L(v)|\ge3$ for every $v\in V(C)-\{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\in V(G)-V(C)$, then $G$ has an $L$-coloring. What happens when $v_1$ and $v_2$ are not adjacent? Then an $L$-coloring need not exist, but in the first paper of this series we have shown that it exists if $|L(v_1)|,|L(v_2)|\ge2$. Here we characterize when an $L$-coloring exists if $|L(v_1)|\ge1$ and $|L(v_2)|\ge2$. This result is a lemma toward a more general theorem along the same lines, which we will use to prove that minimally non-$L$-colorable planar graphs with two precolored cycles of bounded length are of bounded size. The latter result has a number of applications which we pursue elsewhere., Comment: 17 pages, 1 figure

Silbert and Thomas (2013) showed that failures of decisional separability are not, in general, identifiable in fully parameterized $2 \times 2$ Gaussian GRT models. A recent extension of $2 \times 2$ GRT models (GRTwIND) was developed to solve this problem and a conceptually similar problem with the simultaneous identifiability of means and marginal variances in GRT models. Central to the ability of GRTwIND to solve these problems is the assumption of universal perception, which consists of shared perceptual distributions modified by attentional and global scaling parameters (Soto et al., 2015). If universal perception is valid, GRTwIND solves both issues. In this paper, we show that GRTwIND with universal perception and subject-specific failures of decisional separability is mathematically, and thereby empirically, equivalent to a model with decisional separability and failure of universal perception. We then provide a formal proof of the fact that means and marginal variances are not, in general, simultaneously identifiable in $2 \times 2$ GRT models, including GRTwIND. These results can be taken to delineate precisely what the assumption of universal perception must consist of. Based on these results and related recent mathematical developments in the GRT framework, we propose that, in addition to requiring a fixed subset of parameters to determine the location and scale of any given GRT model, some subset of parameters must be set in GRT models to fix the orthogonality of the modeled perceptual dimensions, a central conceptual underpinning of the GRT framework. We conclude with a discussion of perceptual primacy and its relationship to universal perception., Comment: 24 pages, 5 figures, under review at the Journal of Mathematical Psychology