Search Results (18)

We study the controllability of right-invariant bilinear systems on the complex and quaternionic special linear groups $\mathrm{Sl}(n,\mathbb{C})$ and $\mathrm{Sl}(n,\mathbb{H})$. The analysis relies on the Lie saturate$\mathrm{LS}(\Gamma )$, which characterizes controllability through convexity and closure properties of attainable sets, avoiding explicit Lie algebra computations. For $\mathrm{Sl}(n,\mathbb{C})$ with a strongly regular diagonal control matrix, we show that controllability is equivalent to the irreducibility of the drift matrix A, a property verified by the strong connectivity of its associated directed graph. For $\mathrm{Sl}(n,\mathbb{H})$, we derive controllability criteria based on quaternionic entries and the convexity of ${\mathbb{T}}^2$-orbits, which provide efficient sufficient conditions for general n and exact ones in the $2\times 2$ case. These results link algebraic and geometric viewpoints within a unified framework and connect to recent graph-theoretic controllability analyses for bilinear systems on Lie groups. The proposed approach yields constructive and scalable controllability tests for complex and quaternionic systems. Full article

The famous Four-Color Conjecture (now Theorem) states that any planar graph could be colored using four colors. Hadwiger’s conjecture strengthens the Four-Color Conjecture by asserting that every graph with chromatic number t contains a complete minor of order t. In this paper we investigate Hadwiger’s conjecture for the complements of the Petersen graph and the Clebsch graph; both are strongly regular graphs with independence number two (hence dense graphs). We confirm Hajós’ conjecture, hence Hadwiger’s conjecture, for these graphs. Moreover, we show that for each of these graphs the exact hadwiger number is strictly greater its chromatic number. Full article

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erdős, Rényi, and Sós in 1966. The proof relies on a closed-form expression for the Lovász $\vartheta$-function of strongly regular graphs, recently derived by the author. Additionally, this paper considers some known extensions of the theorem, offering discussions that provide insights into the friendship theorem, one of its extensions, and the proposed proof. Leveraging the closed-form expression for the Lovász $\vartheta$-function of strongly regular graphs, this paper further establishes new necessary conditions for a strongly regular graph to be a spanning or induced subgraph of another strongly regular graph. In the case of induced subgraphs, the analysis also incorporates a property of graph energies. Some of these results are extended to regular graphs and their subgraphs. Full article

In this paper, we study self-dual and LCD codes constructed from Kneser graphs K(n, 2) and collinearity graphs of generalized quadrangles using the so-called pure and bordered construction. We determine conditions under which these codes are self-dual or LCD. Further, for the codes over ${\mathbb{Z}}_{2k}$, we give the conditions under which they are Type II. Moreover, we study binary and ternary self-dual and LCD codes from Kneser graphs K(n, 2) and collinearity graphs of generalized quadrangles. Furthermore, from the support designs for certain weights of some of the codes, we construct strongly regular graphs and 3-designs. Full article

In the present work, we classify sets of type (4,n) in PG(3,q). We prove that PG(3,q), apart from the planes of PG(3,3), contains only sets of type (4,n) with standard parameters. Thus, somewhat surprisingly, we conclude that there are no sets of type (4,n) in PG(3,q), q ≠ 3, with non-standard parameters. Full article

Linear codes are the most important family of codes in cryptography and coding theory. Some codes only have a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting $p\equiv 1(mod4)$, we constructed an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index $(p-1)/2$. Their weight distributions were completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal. Full article

In this paper, in the environment of Euclidean Jordan algebras, we establish some inequalities over the Krein parameters of a symmetric association scheme and of a strongly regular graph. Next, we define the modified Krein parameters of a strongly regular graph and establish some admissibility conditions over these parameters. Finally, we introduce some relations over the Krein parameters of a strongly regular graph. Full article

In tag-aware recommender systems, users are strongly encouraged to utilize arbitrary tags to mark items of interest. These user-defined tags can be viewed as a bridge linking users and items. Most tag-aware recommendation models focus on improving the accuracy by introducing ingenious design or complicated structures to handle the tagging information appropriately. Beyond accuracy, diversity is considered to be another important indicator affecting the user satisfaction. Recommending more diverse items will provide more interesting items and commercial sales. Therefore, we propose a diversified tag-aware recommendation model based on graph collaborative filtering. The proposed model establishes a generic graph collaborative filtering framework tailored for tag-aware recommendations. To promote diversity, we adopt two modules: personalized category-boosted negative sampling to select a certain proportion of similar but negative items as negative samples for training, and adversarial learning to make the learned item representation category-free. To improve accuracy, we conduct a two-way TransTag regularization to model the relationship among users, items, and tags. Blending these modules into the proposed framework, we can optimize both the accuracy and diversity concurrently in an end-to-end manner. Experiments on Movielens datasets show that the proposed model can provide diverse recommendations while maintaining a high level of accuracy. Full article

Linear codes with a few weights have been extensively studied due to their wide applications in secret sharing schemes, strongly regular graphs, association schemes, and authentication codes. In this paper, we choose the defining sets from two distinct weakly regular plateaued balanced functions, based on a generic construction of linear codes. Then we construct a family of linear codes with at most five nonzero weights. Their minimality is also examined and the result shows that our codes are helpful in secret sharing schemes. Full article

Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters $(45,22,10,11)$, $(49,24,11,12)$, and $(50,21,8,9)$ that have $Z_6$ as the automorphism group and constructed regular two-graphs from SRGs $(45,22,10,11)$, SRGs $(49,24,11,12)$, and SRGs $(50,21,8,9)$ that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG $(45,22,10,11)$ and 51 new regular two-graphs on 50 vertices leading to 398 new SRG $(49,24,11,12)$. Full article

This paper provides new observations on the Lovász $\theta$-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lovász, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lovász $\theta$-function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lovász $\theta$-function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a k-fold strong power of a regular graph is compared to the Alon–Boppana bound; under a certain condition, the new bound is superior in its exponential growth rate (in k). Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and the Lovász $\theta$-function of each factor. The utility of these bounds is exemplified, leading in some cases to an exact determination of the chromatic numbers of strong products or strong powers of graphs. The present research paper is aimed to have tutorial value as well. Full article

Studies of surface topography including processes of measurement and data analysis have an influence on the description of machined parts with their tribological performance. Usually, surface roughness is analysed when a scale-limited (S-L) surface, excluding short (S-) and length (L-) components from the raw measured data, is defined. Errors in the precise definition of the S-L surface can cause the false estimation of detail properties, especially its tribological performance. Errors can arise when the surface contains some burnished details such as oil pockets, dimples, scratches, or, generally, deep or wide features. The validation of proposed methods for S-L surface definition can also affect the accuracy of the ISO 25178 surface topography parameter calculation. It was found that the application of commonly used procedures, available in commercial software (e.g., least-square fitted cylinder element or polynomial planes, regular or robust Gaussian regression, spline, median or fast Fourier transform filters) can be suitable for precise S-L surface definition. However, some additional analyses, based on power spectral densities, autocorrelation function, texture direction graphs, or spectral characterisation, are strongly required. The effect of the definition of the S-L surface on the values of the ISO 25178 parameters was also comprehensively studied. Some proposals of guidance on how to define an appropriate S-L surface with, respectively, an objective evaluation of surface roughness parameters, were also presented. Full article

Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex that is accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but, in general, they are uncorrelated. Full article

Graph models are found everywhere in natural and human made structures, including process dynamics in physical, biological and social systems. The product of graphs are appropriately used in several combinatorial applications and in the formation of different structural models. In this paper, we present a new product of graphs, namely, maximal product of two vague graphs. Then we describe certain concepts, including strongly, completely, regularity and connectedness on a maximal product of vague graphs. Further, we consider some results of edge regular and totally edge regular in a maximal product of vague graphs. Finally, we present an application for optimization of the biomass based on a maximal product of vague graphs. Full article

We construct several new families of directed strongly regular Cayley graphs (DSRCGs) over the metacyclic group $M_{4n}=⟨a,b|a^n=b^4=1,b^{-1}ab=a^{-1}⟩$ , some of which generalize those earlier constructions. For a prime p and a positive integer $\alpha >1$ , for some cases, we characterize the DSRCGs over $M_{4p^{\alpha }}$ . Full article