Deep learning (DL) identification of vertebral fractures and osteoporosis in lateral spine radiographs and DXA vertebral fracture assessment (VFA) images may improve fracture risk assessment in older adults. In 26 299 lateral spine radiographs from 9276 individuals attending a tertiary-level institution (60% train set; 20% validation set; 20% test set; VERTE-X cohort), DL models were developed to detect prevalent vertebral fracture (pVF) and osteoporosis. The pre-trained DL models from lateral spine radiographs were then fine-tuned in 30% of a DXA VFA dataset (KURE cohort), with performance evaluated in the remaining 70% test set. The area under the receiver operating characteristics curve (AUROC) for DL models to detect pVF and osteoporosis was 0.926 (95% CI 0.908-0.955) and 0.848 (95% CI 0.827-0.869) from VERTE-X spine radiographs, respectively, and 0.924 (95% CI 0.905-0.942) and 0.867 (95% CI 0.853-0.881) from KURE DXA VFA images, respectively. A total of 13.3% and 13.6% of individuals sustained an incident fracture during a median follow-up of 5.4years and 6.4years in the VERTE-X test set (n = 1852) and KURE test set (n = 2456), respectively. Incident fracture risk was significantly greater among individuals with DL-detected vertebral fracture (hazard ratios [HRs] 3.23 [95% CI 2.51-5.17] and 2.11 [95% CI 1.62-2.74] for the VERTE-X and KURE test sets) or DL-detected osteoporosis (HR 2.62 [95% CI 1.90-3.63] and 2.14 [95% CI 1.72-2.66]), which remained significant after adjustment for clinical risk factors and femoral neck bone mineral density. DL scores improved incident fracture discrimination and net benefit when combined with clinical risk factors. In summary, DL-detected pVF and osteoporosis in lateral spine radiographs and DXA VFA images enhanced fracture risk prediction in older adults.

Let $R$ be a ring with identity. The comaximal graph of $R$, denoted by $\Gamma(R)$, is a simple graph with vertex set $R$ and two different vertices $a$ and $b$ are adjacent if and only if $aR+bR=R$. Let $\Gamma_{2}(R)$ be a subgraph of $\Gamma(R)$ induced by $R\backslash\{U(R)\cup J(R)\}$. In this paper, we investigate the genus of the line graph $L(\Gamma(R))$ of $\Gamma(R)$ and the line graph $L(\Gamma_{2}(R))$ of $\Gamma_2(R)$. All finite commutative rings whose genus of $L(\Gamma(R))$ and $L(\Gamma_{2}(R))$ are 0, 1, 2 are completely characterized, respectively.

Geometric deep learning has sparked a rising interest in computer graphics to perform shape understanding tasks, such as shape classification and semantic segmentation. When the input is a polygonal surface, one has to suffer from the irregular mesh structure. Motivated by the geometric spectral theory, we introduce Laplacian2Mesh, a novel and flexible convolutional neural network (CNN) framework for coping with irregular triangle meshes (vertices may have any valence). By mapping the input mesh surface to the multi-dimensional Laplacian-Beltrami space, Laplacian2Mesh enables one to perform shape analysis tasks directly using the mature CNNs, without the need to deal with the irregular connectivity of the mesh structure. We further define a mesh pooling operation such that the receptive field of the network can be expanded while retaining the original vertex set as well as the connections between them. Besides, we introduce a channel-wise self-attention block to learn the individual importance of feature ingredients. Laplacian2Mesh not only decouples the geometry from the irregular connectivity of the mesh structure but also better captures the global features that are central to shape classification and segmentation. Extensive tests on various datasets demonstrate the effectiveness and efficiency of Laplacian2Mesh, particularly in terms of the capability of being vulnerable to noise to fulfill various learning tasks.

The intersection graph of quasinormal subgroups of a group $G$, denoted by $\Gamma_{\mathrm{q}}(G)$, is a graph defined as follows: the vertex set consists of all nontrivial, proper quasinormal subgroups of $G$, and two distinct vertices $H$ and $K$ are adjacent if $H\cap K$ is nontrivial. In this paper, we show that when $G$ is an arbitrary nonsimple group, the diameter of $\Gamma_{\mathrm{q}}(G)$ is in $\{0,1,2,\infty\}$. Besides, all general skew linear groups $\mathrm{GL}_n(D)$ over a division ring $D$ can be classified depending on the diameter of $\Gamma_{\mathrm{q}}(\mathrm{GL}_n(D))$.

We analyze the (parameterized) computational complexity of “fair” variants of bipartite many-to-one matching, where each vertex from the “left” side is matched to exactly one vertex and each vertex from the “right” side may be matched to multiple vertices. We want to find a “fair” matching, in which each vertex from the right side is matched to a “fair” set of vertices. Assuming that each vertex from the left side has one color modeling its “attribute”, we study two fairness criteria. For instance, in one of them, we deem a vertex set fair if for any two colors, the difference between the numbers of their occurrences does not exceed a given threshold. Fairness is, for instance, relevant when finding many-to-one matchings between students and colleges, voters and constituencies, and applicants and firms. Here colors may model sociodemographic attributes, party memberships, and qualifications, respectively. We show that finding a fair many-to-one matching is NP-hard even for three colors and maximum degree five. Our main contribution is the design of fixed-parameter tractable algorithms with respect to the number of vertices on the right side. Our algorithms make use of a variety of techniques including color coding. At the core lie integer linear programs encoding Hall like conditions. We establish the correctness of our integer programs, based on Frank’s separation theorem [Frank, Discrete Math. 1982]. We further obtain complete complexity dichotomies regarding the number of colors and the maximum degree of each side.

An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the 5-regular case and show that among the subgraphs of minimum degree at least 3 of 5-regular graphs, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all 5-regular Abelian Cayley graphs which do not have an internal partition.

The comparison of multiple (labeled) graphs with unrelated vertex sets is an important task in diverse areas of applications. Conceptually, it is often closely related to multiple sequence alignments since one aims to determine a correspondence, or more precisely, a multipartite matching between the vertex sets. There, the goal is to match vertices that are similar in terms of labels and local neighborhoods. Alignments of sequences and ordered forests, however, have a second aspect that does not seem to be considered for graph comparison, namely the idea that an alignment is a superobject from which the constituent input objects can be recovered faithfully as well-defined projections. Progressive alignment algorithms are based on the idea of computing multiple alignments as a pairwise alignment of the alignments of two disjoint subsets of the input objects. Our formal framework guarantees that alignments have compositional properties that make alignments of alignments well-defined. The various similarity-based graph matching constructions do not share this property and solve substantially different optimization problems. We demonstrate that optimal multiple graph alignments can be approximated well by means of progressive alignment schemes. The solution of the pairwise alignment problem is reduced formally to computing maximal common induced subgraphs. Similar to the ambiguities arising from consecutive indels, pairwise alignments of graph alignments require the consideration of ambiguous edges that may appear between alignment columns with complementary gap patterns. We report a simple reference implementation in Python/NetworkX intended to serve as starting point for further developments. The computational feasibility of our approach is demonstrated on test sets of small graphs that mimimc in particular applications to molecular graphs.

Let $G$ be a simple graph or a multigraph. The vertex connectivity $\kappa(G)$ of $G$ is the minimum size of a vertex set $S$ such that $G-S$ is disconnected or has only one vertex. We denote by $\lambda_{3}(G)$ the third largest eigenvalue of the adjacency matrix of $G$. In this paper, we present an upper bound for $\lambda_{3}(G)$ in a $d$-regular (multi-)graph $G$ which guarantees that $\kappa(G)\geq t+1$, which is based on the result of Abiad et al. [Spectral bounds for the connectivity of regular graphs with given order. Electron. J. Linear Algebra 34:428-443, 2018]. Furthermore, we improve the upper bound for $\lambda_{3}(G)$ in a $d$-regular multigraph which assures that $\kappa(G)\geq 2$.

A bijective product square [Formula: see text]-cordial labeling of a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text] is a bijection [Formula: see text] from [Formula: see text] to [Formula: see text] such that the induced edge function [Formula: see text] from [Formula: see text] to [Formula: see text] defined as [Formula: see text] for every edge [Formula: see text] satisfies the following condition. (i) If [Formula: see text] is the number of edges labeled with [Formula: see text] under [Formula: see text], then [Formula: see text] where [Formula: see text]. A graph which admits a bijective product square [Formula: see text]-cordial labeling is called bijective product square [Formula: see text]-cordial graph. In this paper, we find upper limit of the size of the bijective product square [Formula: see text]-cordial graph for [Formula: see text] and [Formula: see text]. Also, we prove that every graph is a subgraph of a connected bijective product square [Formula: see text]-cordial graph for all odd prime [Formula: see text]. In addition, we establish the relation between bijective product square [Formula: see text]-cordial and [Formula: see text]-product cordial graphs.

The concept of [Formula: see text]-independent set, introduced by Fink and Jacobson in 1986, is a natural generalization of classical independence set. A [Formula: see text]-independent set is a set of vertices whose induced subgraph has maximum degree at most [Formula: see text]. The [Formula: see text]-independence number of [Formula: see text], denoted by [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-independent set of [Formula: see text]. As a natural counterpart of the [Formula: see text]-independence number, we introduced the concept of [Formula: see text]-edge-independence number. An edge set [Formula: see text] in [Formula: see text] is called [Formula: see text]-edge-independent if the maximum degree of the subgraph induced by the edges in [Formula: see text] is less or equal to [Formula: see text]. The [Formula: see text]-edge-independence number, denoted [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-edge-independent set. In this paper, we study the Nordhaus–Gaddum-type results for the parameter [Formula: see text] and [Formula: see text]. We obtain sharp upper and lower bounds of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. Some graph classes attaining these bounds are also given.

For a graph [Formula: see text] and positive integers [Formula: see text], [Formula: see text], a [Formula: see text]-tree-vertex coloring of [Formula: see text] refers to a [Formula: see text]-vertex coloring of [Formula: see text] satisfying every component of each induced subgraph generated by every set of vertices with the same color forms a tree with maximum degree not larger than [Formula: see text], and it is called equitable if the difference between the cardinalities of every pair of sets of vertices with the same color is at most [Formula: see text]. The strong equitable vertex [Formula: see text]-arboricity of [Formula: see text], denoted by [Formula: see text], is defined as the least positive integer [Formula: see text] satisfying [Formula: see text], which admits an equitable [Formula: see text]-tree-vertex coloring for each integer [Formula: see text] with [Formula: see text]. The strong equitable vertex [Formula: see text]-arboricity of a graph is very useful in graph theory applications such as load balance in parallel memory systems, constructing timetables and scheduling. In this paper, we present the tight upper and lower bounds on [Formula: see text] for an arbitrary graph [Formula: see text] with [Formula: see text] vertices and a given integer [Formula: see text] with [Formula: see text], and we characterize the extremal graphs [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text], respectively. Based on the above extremal results, we further obtain the Nordhaus–Gaddum-type results for [Formula: see text] of graphs [Formula: see text] with [Formula: see text] vertices for a given integer [Formula: see text] with [Formula: see text].

Dominating David-derived networks are widely studied due to their fractal nature, with applications in topology, chemistry, and computer sciences. The use of molecular structure descriptors is a standard procedure that is used to correlate the biological activity of molecules with their chemical structures, which can be useful in the field of pharmacology. This article's goal is to develop analytically closed computing formulas for eccentricitybased descriptors of the second type of dominating David-derived network. Thermodynamic characteristics, physicochemical properties, and chemical and biological activities of chemical graphs are just a few of the many properties that may be determined using these computation formulas. Vertex sets were initially divided according to their degrees, eccentricities, and cardinalities of occurrence. The eccentricity-based indices are then computed using some combinatorics and these partitions. Total eccentricity, average eccentricity, and the Zagreb index are distance-based topological indices utilized in this study for the second type of dominating David-derived network, denoted as D2(m). These calculations will assist the readers in estimating the fractal and difficult-tohandle thermodynamic and physicochemical aspects of chemical structure. Apart from configuration and impact resistance, the D2(m) design has been used for fundamental reasons in a variety of technical and scientific advancements.

This article investigates robust stabilizing control of biological systems modeled by Boolean networks (BNs). A population of BNs is considered where a majority of BNs have the same BN dynamics, but some BNs are inflicted by mutations damaging particular nodes, leading to perturbed dynamics that prohibit global stabilization to the desired attractor. The proposed control strategy consists of two steps. First, the nominal BN is transformed and curtailed into a sub-BN via a simple coordinate transformation and network reduction associated with the desired attractor. The feedback vertex set (FVS) control is then applied to the reduced BN to determine the control inputs for the nominal BN. Next, the control inputs derived in the first step and mutated nodes are applied to the nominal BN so as to identify residual dynamics of perturbed BNs, and additional control inputs are selected according to the canalization effect of each node. The overall control inputs are applied to the BN population, so that the nominal BN converges to the desired attractor and perturbed BNs to their own attractors that are the closest possible to the desired attractor. The performance of the proposed robust control scheme is validated through numerical experiments on random BNs and a complex biological network.

The exploration of edge metric dimension and its applications has been an ongoing discussion, particularly in the context of nanosheet graphs formed from the octagonal grid. Edge metric dimension is a concept that involves uniquely identifying the entire edge set of a structure with a selected subset from the vertex set, known as the edge resolving set. Let's consider two distinct edge resolving sets, denoted as Re1 and Re2, where Re1≠Re2. In such instances, it indicates that the graph G possesses a double-edge resolving set. This implies the existence of two different subsets of the vertex set, each capable of uniquely identifying the entire edge set of the graph. In this article, we delve into the edge metric dimension of nanosheet graphs derived from the octagonal grid. Additionally, we initiate a discussion on the exchange property associated with the edge resolving set. The exchange property holds significance in the study of resolving sets, playing a crucial role in comprehending the structure and properties of the underlying graph.

Reliability of [formula omitted]-ary [formula omitted]-dimensional hypercubes under embedded restriction

The Randić-type graph invariants are extensively investigated vertex-degree-based topological indices and have gained much prominence in recent years. The general Randić and zeroth-order general Randić indices are Randić-type graph invariants and are defined for a graph G with vertex set V as RαG=∑υi∼υjdidjα and QαG=∑vi∈Vdiα, respectively, where α is an arbitrary real number, di denotes the degree of a vertex υi, and υi∼υj represents the adjacency of vertices υi and υj in G. Establishing relationships between two topological indices holds significant importance for researchers. Some implicit inequality relationships between Rα and Qα have been derived so far. In this paper, we establish explicit inequality relationships between Rα and Qα. Also, we determine linear inequality relationships between these graph invariants. Moreover, we obtain some new inequalities for various vertex-degree-based topological indices by the appropriate choice of α.

We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm. Funding: This work was supported by the Engineering and Physical Sciences Research Council [Grant EP/V002813/1]; the National Science Foundation [Grants DMS-1763817, DMS-2120644, and DMS-2303251]; and Alexander von Humboldt-Stiftung.

We study the distance on the Bruhat–Tits building of the group [Formula: see text] (and its other combinatorial properties). Coding its vertices by certain matrix representatives, we introduce a way how to build formulas with combinatorial meanings. In Theorem 1, we give an explicit formula for the graph distance [Formula: see text] of two vertices [Formula: see text] and [Formula: see text] (without having to specify their common apartment). Our main result, Theorem 2, then extends the distance formula to a formula for the smallest total distance of a vertex from a given finite set of vertices. In the appendix we consider the case of [Formula: see text] and give a formula for the number of edges shared by two given apartments.

This article has been retracted. A retraction notice can be found at https://doi.org/10.3233/JIFS-219433.

Given a non-decreasing sequence S=(s1,s2,…,sk) of positive integers, an S-packing coloring of a graph G is a partition of the vertex set of G into k subsets {V1,V2,…,Vk} such that for each 1≤i≤k, the distance between any two distinct vertices u and v in Vi is at least si+1. In this paper, we study the problem of S-packing coloring of cubic Halin graphs, and we prove that every cubic Halin graph is (1,1,2,3)-packing colorable. In addition, we prove that such graphs are (1,2,2,2,2,2)-packing colorable.