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We give a complete characterisation of the cubic graphs with no eigenvalues in the open interval $(-1,1)$. There are two infinite families, one due to Guo and Mohar [Linear Algebra Appl. 449:68--75] the other due to Koll\'ar and Sarnak [Communications of the AMS. 1,1--38], and $14$ "sporadic" graphs on at most $32$ vertices. This allows us to show that $(-1,1)$ is a maximal spectral gap set for cubic graphs. Our techniques including examination of various substructure and an application of the classification of generalized line graphs.

Mitotic activity is an important feature for grading several cancer types. Counting mitotic figures (MFs) is a time-consuming, laborious task prone to inter-observer variation. Inaccurate recognition of MFs can lead to incorrect grading and hence potential suboptimal treatment. In this study, we propose an artificial intelligence (AI)-aided approach to detect MFs in digitised haematoxylin and eosin-stained whole slide images (WSIs). Advances in this area are hampered by the limited number and types of cancer datasets of MFs. Here we establish the largest pan-cancer dataset of mitotic figures by combining an in-house dataset of soft tissue tumours (STMF) with five open-source mitotic datasets comprising multiple human cancers and canine specimens (ICPR, TUPAC, CCMCT, CMC and MIDOG++). This new dataset identifies 74,620 MFs and 105,538 mitotic-like figures. We then employed a two-stage framework (the Optimised Mitoses Generator Network (OMG-Net) to classify MFs. The framework first deploys the Segment Anything Model (SAM) to automate the contouring of MFs and surrounding objects. An adapted ResNet18 is subsequently trained to classify MFs. OMG-Net reaches an F1-score of 0.84 on pan-cancer MF detection (breast carcinoma, neuroendocrine tumour and melanoma), largely outperforming the previous state-of-the-art MIDOG++ benchmark model on its hold-out testing set (e.g. +16% F1-score on breast cancer detection, p<0.001) thereby providing superior accuracy in detecting MFs on various types of tumours obtained with different scanners.

There are finitely many graphs with diameter $2$ and girth 5. What if the girth 5 assumption is relaxed? Apart from stars, are there finitely many triangle-free graphs with diameter $2$ and no $K_{2,3}$ subgraph? This question is related to the existence of triangle-free strongly regular graphs, but allowing for a range of co-degrees gives the question a more extremal flavour. More generally, for fixed $s$ and $t$, are there infinitely many twin-free triangle-free $K_{s,t}$-free graphs with diameter 2? This paper presents partial results regarding these questions, including computational results, potential Cayley-graph and probabilistic constructions.

For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of \emph{synchronising} and \emph{separating} are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of \emph{spreading} permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an \emph{AB-Lemma}) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups.

We present the first experimental observation of K$_{\beta}$ emission from highly charged Mg ions at solid density, driven by intense x-rays from a free electron laser. The presence of K$_{\beta}$ emission indicates the $n=3$ atomic shell is relocalized for high charge states, providing an upper constraint on the depression of the ionization potential. We explore the process of state relocalization in dense plasmas from first principles using finite-temperature density functional theory alongside a wavefunction localization metric, and find excellent agreement with experimental results., Comment: 22 pages, 13 figures

Point counts (PCs) are widely used in biodiversity surveys, but despite numerous advantages, simple PCs suffer from several problems: detectability, and therefore abundance, is unknown; systematic spatiotemporal variation in detectability produces biased inferences, and unknown survey area prevents formal density estimation and scaling-up to the landscape level. We introduce integrated distance sampling (IDS) models that combine distance sampling (DS) with simple PC or detection/nondetection (DND) data and capitalize on the strengths and mitigate the weaknesses of each data type. Key to IDS models is the view of simple PC and DND data as aggregations of latent DS surveys that observe the same underlying density process. This enables estimation of separate detection functions, along with distinct covariate effects, for all data types. Additional information from repeat or time-removal surveys, or variable survey duration, enables separate estimation of the availability and perceptibility components of detectability. IDS models reconcile spatial and temporal mismatches among data sets and solve the above-mentioned problems of simple PC and DND data. To fit IDS models, we provide JAGS code and the new IDS() function in the R package unmarked. Extant citizen-science data generally lack adjustments for detection biases, but IDS models address this shortcoming, thus greatly extending the utility and reach of these data. In addition, they enable formal density estimation in hybrid designs, which efficiently combine distance sampling with distance-free, point-based PC or DND surveys. We believe that IDS models have considerable scope in ecology, management, and monitoring., Comment: 24 pages, plus 5 figures, plus 2 Appendices

We classify, up to some notoriously hard cases, the rank 3 graphs which fail to meet either the Delsarte or the Hoffman bound. As a consequence, we resolve the question of separation for the corresponding rank 3 primitive groups and give new examples of synchronising, but not $\mathbb{Q}\mathrm{I}$, groups of affine type., Comment: Added acknowledgements and noted the third author's change of institution

Realizing the full potential of ultrahigh-intensity lasers for particle and radiation generation will require multi-beam arrangements due to technology limitations. Here, we investigate how to optimize their coupling with solid targets. Experimentally, we show that overlapping two intense lasers in a mirror-like configuration onto a solid with a large preplasma can greatly improve the generation of hot electrons at the target front and ion acceleration at the target backside. The underlying mechanisms are analyzed through multidimensional particle-in-cell simulations, revealing that the self-induced magnetic fields driven by the two laser beams at the target front are susceptible to reconnection, which is one possible mechanism to boost electron energization. In addition, the resistive magnetic field generated during the transport of the hot electrons in the target bulk tends to improve their collimation. Our simulations also indicate that such effects can be further enhanced by overlapping more than two laser beams.

This paper characterizes the actual science performance of the James Webb Space Telescope (JWST), as determined from the six month commissioning period. We summarize the performance of the spacecraft, telescope, science instruments, and ground system, with an emphasis on differences from pre-launch expectations. Commissioning has made clear that JWST is fully capable of achieving the discoveries for which it was built. Moreover, almost across the board, the science performance of JWST is better than expected; in most cases, JWST will go deeper faster than expected. The telescope and instrument suite have demonstrated the sensitivity, stability, image quality, and spectral range that are necessary to transform our understanding of the cosmos through observations spanning from near-earth asteroids to the most distant galaxies., Comment: 5th version as accepted to PASP; 31 pages, 18 figures; https://iopscience.iop.org/article/10.1088/1538-3873/acb293

A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise. Pontus von Br\"omssen (n\'e Andersson) showed that the existence of such an automorphism is independent of the orientation, and considered the question of counting pairwise non-isomorphic even graphs. Based on computational evidence, he made the rather surprising conjecture that the number of pairwise non-isomorphic even graphs on $n$ vertices is equal to the number of pairwise non-isomorphic tournaments on $n$ vertices. We prove this conjecture using a counting argument with several applications of the Cauchy-Frobenius Theorem., Comment: 9 pages. Corrected minor non-mathematical typos and slightly expanded the introduction

Every synchronising permutation group is primitive and of one of three types: affine, almost simple, or diagonal. We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that $\mathrm{PSL}(2,q)\times \mathrm{PSL}(2,q)$ acting in its diagonal action on $\mathrm{PSL}(2,q)$ is separating, and hence synchronising, for $q=13$ and $q=17$. Furthermore, we show that such groups are non-spreading for all prime powers $q$.

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the $195\,826\,352$ conjugacy classes of transitive subgroups of the symmetric group $S_{48}$ of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive graphs of degree $48$, of which there are $1\,538\,868\,366$, all but $0.1625\%$ of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published by the third author as a separate paper

We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if $N$ is an excluded minor of rank at least ten, then $N$ is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that $N$ already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows., Comment: Published version; Advances in Combinatorics

We say that a finite group $G$ acting on a set $\Omega$ has Property $(*)_p$ for a prime $p$ if $P_\omega$ is a Sylow $p$-subgroup of $G_\omega$ for all $\omega\in\Omega$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ arose in the recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozes groups, and he determined the values of $p$ for which the alternating group $A_n$ and symmetric group $S_n$ acting on $n$ points has Property $(*)_p$. In this paper, we extend this result to finite $2$-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy Property $(*)_p$ for an allowable prime $p$., Comment: minor corrections

Dose painting has been developed to modulate the required dose in the target area without increasing the toxicity in healthy areas. Apart from determining the accurate location and size of tumors, quantitative functional imaging can be used to implement the dose painting. Functional imaging, such as multi-parameter MRI and PET CT, allows us to achieve biological dose escalation by increasing the dose in certain areas or voxels that are therapy-resistant in the gross tumor volume while reducing the dose in the less aggressive area or voxels. Functional imaging can serve as an indicator of therapeutic intervention in radiotherapy due to microscopic tissue properties. With such biological indicators, the personalized radiation dose can be tailored to a specific contour or a voxel using dose painting. In this review, we firstly discuss several quantitative functional imaging techniques including PET-CT and multi-parameter MRI. Furthermore, theoretical and experimental comparisons for dose painting by contours (DPBC) and dose painting by numbers (DPBN), along with outcome analysis after dose painting are provided. Finally, we conclude major challenges and future directions in this field through which we hope to inspire exciting developments and fruitful research avenues.

We present picosecond-resolution streaked K-shell spectra from 400 nm-diameter nickel nanowire arrays, demonstrating the ability to generate large volumes of high energy density plasma when combined with the longer pulses typical of the largest short pulse lasers. After irradiating the wire array with 100 J, 600 fs ultra-high-contrast laser pulses focussed to $>10^{20}$ W/cm$^{2}$ at the Orion laser facility, we combine atomic kinetics modeling of the streaked spectra with 2D collisional particle-in-cell simulations to describe the evolution of material conditions within these samples for the first time. We observe a three-fold enhancement of helium-like emission compared to a flat foil in a near-solid-density plasma sustaining keV temperatures for tens of picoseconds, the result of strong electric return currents heating the wires and causing them to explode and collide., Comment: 5 pages, 5 figures

In this paper, we study the flip graph on the perfect matchings of a complete graph of even order. We investigate its combinatorial and spectral properties including connections to the signed reversal graph and we improve a previous upper bound on its chromatic number., Comment: 15 pages, 6 figures, 2 tables

We examine the subgroup $D(G)$ of a transitive permutation group $G$ which is generated by the derangements in $G$. Our main results bound the index of this subgroup: we conjecture that, if $G$ has degree $n$ and is not a Frobenius group, then $|G:D(G)|\leqslant\sqrt{n}-1$; we prove this except when $G$ is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding $|H:R(H)|$, where $H$ is a linear group on a finite vector space and $R(H)$ is the subgroup of $H$ generated by elements having eigenvalue~$1$. If $G$ is a Frobenius group, then $D(G)$ is the Frobenius kernel, and so $G/D(G)$ is isomorphic to a Frobenius complement. We give some examples where $D(G)\ne G$, and examine the group-theoretic structure of $G/D(G)$; in particular, we construct groups $G$ in which $G/D(G)$ is not a Frobenius complement.

The development of high-brightness free-electron lasers (FEL) has revolutionised our ability to create and study matter in the high-energy-density (HED) regime. Current diagnostic techniques have been very successful in yielding information on fundamental thermodynamic plasma properties, but provide only limited or indirect information on the detailed quantum structure of these systems, and on how it is affected by ionization dynamics. Here we show how the electronic structure of solid-density nickel, heated to temperatures of 10's of eV on femtosecond timescales, can be studied by resonant (Raman) inelastic x-ray scattering (RIXS) using the Linac Coherent Light Source FEL. We present single-shot measurements of the valence density of states in the x-ray-heated transient system, and extract simultaneously electron temperatures, ionization, and ionization potential energies. The RIXS spectrum provides a wealth of information on the valence structure of the HED system that goes beyond what can be extracted from x-ray absorption or emission spectroscopy alone.

We explore the interplay between algebraic combinatorics and algorithmic problems in graph theory by defining a polynomial with connections to correspondence colouring (also known as DP-colouring), a recent generalization of list-colouring, and the Unique Games Conjecture. Like the chromatic polynomial of a graph, we are able to evaluate this polynomial at a point, despite the complexity of computing this polynomial. We construct a cover of a graph $X$ by blowing up each vertex to a set of $r$ vertices and joining each pair of sets corresponding to adjacent vertices by a matching with $r$ edges. To each cover $Y$ of $X$ we associate a polynomial $\xi(Y,t)$, called the transversal polynomial. The coefficient $t^k$ of $\xi(Y,t)$ is the number of $k$-edge induced subgraphs of $Y$ whose vertex set is a transversal of the set system given by the blown-up vertices. We show that $\xi(Y,t)$ satisfies a contraction-deletion formula, and that if $n=|V_X|$ and the cover has index $r$, then $\xi(Y,-(r-1)) \equiv 0 \mod r^n$., Comment: 15 pages, 4 figures

In this paper we consider the number of Hamilton cycles in planar cubic graphs of high cyclic edge-connectivity, answering two questions raised by Chia and Thomassen ("On the number of longest and almost longest cycles in cubic graphs", Ars Combin., 104, 307--320, 2012) about extremal graphs in these families. In particular, we find families of cyclically $5$-edge connected planar cubic graphs with more Hamilton cycles than the generalized Petersen graphs $P(2n,2)$. The graphs themselves are fullerene graphs that correspond to certain carbon molecules known as nanotubes --- more precisely, the family consists of the zigzag nanotubes of (fixed) width $5$ and increasing length. In order to count the Hamilton cycles in the nanotubes, we develop methods inspired by the transfer matrices of statistical physics. We outline how these methods can be adapted to count the Hamilton cycles in nanotubes of greater (but still fixed) width, with the caveat that the resulting expressions involve matrix powers. We also consider cyclically $4$-edge-connected cubic planar graphs with few Hamilton cycles, and exhibit an infinite family of such graphs each with exactly $4$ Hamilton cycles. Finally we consider the "other extreme" for these two classes of graphs, thus investigating cyclically $4$-edge connected cubic planar graphs with many Hamilton cycles and the cyclically $5$-edge connected cubic planar graphs with few Hamilton cycles. In each of these cases, we present partial results, examples and conjectures regarding the graphs with few or many Hamilton cycles.

We describe two similar but independently-coded computations used to construct a complete catalogue of the transitive groups of degree less than $48$, thereby verifying, unifying and extending the catalogues previously available. From this list, we construct all the vertex-transitive graphs of order less than $48$. We then present a variety of summary data regarding the transitive groups and vertex-transitive graphs, focussing on properties that seem to occur most frequently in the study of groups acting on graphs. We illustrate how such catalogues can be used, first by finding a complete list of the elusive groups of order at most $47$ and then by completely determining which groups of order at most $47$ are CI groups

A graph is $\textit{cubelike}$ if it is a Cayley graph for some elementary abelian $2$-group $\mathbb{Z}_2^n$. The core of a graph is its smallest subgraph to which it admits a homomorphism. More than ten years ago, Ne\v{s}et\v{r}il and \v{S}\'amal (On tension-continuous mappings. $\textit{European J. Combin.,}$ 29(4):1025--1054, 2008) asked whether the core of a cubelike graph is cubelike, but since then very little progress has been made towards resolving the question. Here we investigate the structure of the core of a cubelike graph, deducing a variety of structural, spectral and group-theoretical properties that the core "inherits" from the host cubelike graph. These properties constrain the structure of the core quite severely --- even if the core of a cubelike graph is not actually cubelike, it must bear a very close resemblance to a cubelike graph. Moreover we prove the much stronger result that not only are these properties inherited by the core of a cubelike graph, but also by the orbital graphs of the core. Even though the core and its orbital graphs look very much like cubelike graphs, we are unable to show that this is sufficient to characterise cubelike graphs. However, our results are strong enough to eliminate all non-cubelike vertex-transitive graphs on up to $32$ vertices as potential cores of cubelike graphs (of any size). Thus, if one exists at all, a cubelike graph with a non-cubelike core has at least $128$ vertices and its core has at least $64$ vertices., Comment: 27 pages

By the construction of suitable graphs and the determination of their chromatic polynomials, we resolve two open questions concerning real chromatic roots. First we exhibit graphs for which the Beraha number $B_{10} = (5 + \sqrt{5})/2$ is a chromatic root. As it was previously known that no other non-integer Beraha number is a chromatic root, this completes the determination of precisely which Beraha numbers can be chromatic roots. Next we construct an infinite family of $3$-connected graphs such that for any $k \geqslant 1$, there is a member of the family with $q=2$ as a chromatic root of multiplicity at least $k$. The former resolves a question of Salas and Sokal [J. Statist. Pys. 104 (2001) pp. 609--699] and the latter a question of Dong and Koh [J. Graph Theory 70 (2012) pp. 262--283]., Comment: This first section of this paper was previously uploaded as arxiv:1611.01941

A pair $(A,B)$ of square $(0,1)$-matrices is called a \emph{Lehman pair} if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called \emph{Lehman matrices}. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman matrices with $k=-1$ are essentially equivalent to \emph{partitionable graphs} (also known as $(\alpha,\omega)$-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, we view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite graph, focusing in particular on the case where the graph is cubic. From this perspective, we identify two constructions that generate cubic Lehman graphs from smaller Lehman graphs. The most prolific of these constructions involves repeatedly replacing suitable pairs of edges with a particular $6$-vertex subgraph that we call a $3$-rung ladder segment. Two decades ago, L\"{u}tolf \& Margot initiated a computational study of mni matrices and constructed a catalogue containing (among other things) a listing of all cubic Lehman matrices with $k =1$ of order up to $17 \times 17$. We verify their catalogue (which has just one omission), and extend the computational results to $20 \times 20$ matrices. Of the $908$ cubic Lehman matrices (with $k=1$) of order up to $20 \times 20$, only two do not arise from our $3$-rung ladder construction. However these exceptions can be derived from our second construction, and so our two constructions cover all known cubic Lehman matrices with $k=1$.