This study is motivated by the need to investigate the largest possible collection of classical separation axioms within a newly introduced triple structure of generalized primal topological spaces, and to understand how primal collections influence these familiar notions. The purpose of the paper is to extend several classical concepts by introducing new classes of separation axioms, including (g,P)-Di, (g,P)-Ti, and (g,P)-Ri for i=0,1,2. Within the same framework, we also define (g,P)-Gδ sets and (g,P)-Fσ sets, which naturally lead to new symmetric variants of separation axioms such as (g,P)-Rδ, (g,P)-weakly regular, (g,P)-RDδ, and (g,P)-RD★. The main contribution of this work lies in establishing the relationships among these newly introduced axioms and demonstrating how primal collections affect their behavior. Several illustrative examples based on simple graphs are provided to highlight the structure and significance of the results. Overall, the findings offer a broader perspective on separation phenomena in generalized primal settings and deepen the understanding of symmetry within these spaces.

Graph theory plays a central role in mathematics, biology, chemistry, computer science, and related disciplines. It has many applications in everyday life, particularly in chemistry, biology, and network theory. Chemical graph theory is a subfield devoted to the mathematical representation and analysis of molecular structures. It is also used in the calculation of topological indices and the prediction of many chemical properties. A topological index is a numerical parameter that characterizes the molecular structure based on its corresponding molecular graph. Consider a simple molecular graph G = (V(G), E(G)) with no loops, multiple edges, or directed edges. Numerous topological indices have been defined and studied for molecular graphs. The vertex and edge eccentric connectivity indices, along with their modified versions, play a significant role in QSPR/QSAR studies within the framework of chemical graph theory. Recently, various studies have been conducted on the backbone DNA graphs. The repeating cycles in the backbone DNA graphs indicate that the graph possesses a periodic and regular symmetry. This symmetry is taken into account in deriving closed formulas for topological index values such as the eccentric connectivity indices. In this paper, some eccentric connectivity indices based on vertices and edges of backbone DNA graphs DNAn have been computed. Furthermore, the two-dimensional plots of DNAn were generated using Cartesian coordinates.

A theta curve is a spatial embedding of the θ-graph in the threesphere, taken up to ambient isotopy. We define the determinant of a theta curve as an integer-valued invariant arising from the first homology of its Klein cover. When a theta curve is simple, containing a constituent unknot, we prove that the determinant of the theta curve is the product of the determinants of the constituent knots. Our proofs are combinatorial, relying on Kirchhoff’s Matrix Tree Theorem and spanning tree enumeration results for symmetric, signed, planar graphs.

Let G be a simple graph and k\in\mathbb{N} . Two important graph operations are: the k -th graph power of G , denoted by G^{k} , where G^{k} is the graph obtained from G by adding an edge between every pair of vertices that have a distance at most k , and the complement graph of G , denoted by \overline{G} . In this paper, we studied the relation between the independence numbers of \overline{G}^{k} and \overline{G^k} .

We characterize twisted right-angled Artin groups (T-RAAGs) that are subgroup separable using only their defining mixed graphs: such a group is subgroup separable if and only if the underlying simplicial graph contains no induced path or square on four vertices. This generalizes the results of Metaftsis and Raptis on classical right-angled Artin groups. We also show that the subgroup membership problem is decidable when the group is coherent—which occurs precisely when the defining mixed graph is chordal. Furthermore, we exhibit a family of cone-type mixed graphs for which the corresponding T-RAAGs have decidable rational and submonoid membership problems.

Detachable pairs in 3-connected matroids and simple 3-connected graphs

The delivery of digital information in Indonesian-language news presents challenges in efficiently capturing the core information. This study proposes a combination of the TextRank algorithm and a simple Graph Neural Network (GNN) to improve the quality of automatic text summarization. TextRank is used to construct a sentence graph based on TF-IDF similarity and cosine similarity, followed by training a SimpleGNN model to optimize sentence scores. Evaluations were conducted on 1,000 articles from the Liputan6 dataset using the ROUGE metric (ROUGE-1, ROUGE-2, and ROUGE-L). The results show that this combined method improves performance compared to pure TextRank, especially in capturing semantic relationships between sentences. This study demonstrates that the integration of a simple GNN can enrich representations in graphs and provide more informative and contextual summaries.

An infinite family of simple graphs underlying chiral, orientable reflexible and non-orientable rotary maps

This paper aims to associate a new graph to nonzero unital modules over commutative rings. Let $R$ be a commutative ring having a nonzero identity and $M$ be a nonzero unital $R$-module. The zero intersection graph of annihilator ideals of $R$-module $M$, denoted by $\mathfrak{C}_{R}(M)$, is a simple (undirected) graph whose vertex set $M^{\star}=M-\{0\},\ $and two distinct vertices $m$ and $m^{\prime}$ are adjacent if $ann_{R}(m)\cap ann_{R}(m^{\prime})=(0).$\ We investigate the conditions under which $\mathfrak{C}_{R}(M)$ is a star graph, bipartite graph, complete graph, edgeless graph. Furthermore, we characterize certain classes of modules and rings such as torsion-free modules, torsion modules, semisimple modules, quasi-regular rings, and modules satisfying Property $T$ in terms of their graphical properties.

We investigate the fundamental trade-off between entropy and the Gini index within income distributions, employing a stochastic framework to expose deficiencies in conventional inequality metrics. Anchored in the principle of maximum entropy (ME), we position entropy as a key marker of societal robustness, while the Gini index, identical to the (second-order) K-spread coefficient, captures spread but neglects dynamics in distribution tails. We recommend supplanting Lorenz profiles with simpler graphs such as the odds and probability density functions, and a core set of numerical indicators (K-spread K₂/μ, standardized entropy Φμ, and upper and lower tail indices, ξ, ζ) for deeper diagnostics. This approach fuses ME into disparity evaluation, highlighting a path to harmonize fairness with structural endurance. Drawing from percentile records in the World Income Inequality Database from 1947 to 2023, we fit flexible models (Pareto–Burr–Feller, Dagum) and extract K-moments and tail indices. The results unveil a concave frontier: moderate Gini reductions have little effect on entropy, but aggressive equalization incurs steep stability costs. Country-level analyses (Argentina, Brazil, South Africa, Bulgaria) link entropy declines to political ruptures, positioning low entropy as a precursor to instability. On the other hand, analyses based on the core set of indicators for present-day geopolitical powers show that they are positioned in a high stability area.

We investigate a generalization of the bondage number of a graph called the k-synchronous bondage number. The k-synchronous bondage number of a graph is the smallest number of edges that, when removed, increases the domination number by k. In this paper, we discuss the 2-synchronous bondage number and then generalize to the k -synchronous bondage number. We present k -synchronous bondage numbers for several graph classes and give bounds for general graphs. We propose this characteristic as a metric of the connectivity of a simple graph with possible uses in the field of network design and optimization.

The non-commuting conjugacy class graph (abbreviated as NCCC-graph) of a finite non-abelian group $H$ is a simple undirected graph whose vertex set is the set of conjugacy classes of non-central elements of $H$ and two vertices, $a^H$ and $b^H$ are adjacent if $a'b' \ne b'a'$ for all $a' \in a^H$ and $b' \in b^H$. In this paper, we compute distance spectrum, distance Laplacian spectrum, distance signless Laplacian spectrum along with their respective energies and Wiener index of NCCC-graphs of $H$ when the central quotient of $H$ is isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ (for any prime $p$) or $D_{2n}$ (for any integer $n \geq 3$). As a consequence, we compute various distance spectra, energies and Wiener index of NCCC-graphs of the dihedral group, dicyclic group, semidihedral group along with the groups $U_{(n,m)}$, $U_{6n}$ and $V_{8n}$. Thus we obtain sequences of positive integers that can be realized as Wiener index of NCCC-graphs of certain groups. In particular, we solve Inverse Wiener index Problem for NCCC-graphs of groups when $n$ is a perfect square. We further characterize the above-mentioned groups such that their NCCC-graphs are D-integral, DL-integral and DQ-integral. We also compare various distance energies of NCCC-graphs of the above mentioned groups and characterize those groups subject to the inequalities involving various distance energies.

The Albertson irregularity measure is defined as $Alb(\Gamma)=\sum_{uv\in E(\Gamma)} \vert d(u)-d(v)\vert.$ In this work, the concept of Albertson energy is extended from simple graphs to graphs with self-loops. Also the expression for the Albertson eigenvalues of a graph with self-loops are given. Some bounds on the Albertson energy of graphs with self-loops and the spread of $Alb(\Gamma_S)$ are obtained. In the last section, the Albertson energy of complete, complete bipartite, crown and thorn graphs with self-loops are computed.

Absorption, distribution, metabolism, and excretion (ADME) properties are among the key factors in determining the success of lead discovery and optimization campaigns. Fast and accurate prediction of molecular ADME profiles is hence of particular interest as a prioritization tool before costly experimental assays. However, the severe scarcity of publicly available training data for ADME prediction has hindered the development of improved machine learning models. Recently, industry teams have taken the important step to release the predicted labels from their in-house trained models for public domain chemical structures. In this paper, leveraging these large and diverse surrogate data sets, we propose the adoption of transfer learning using a simple multitask graph neural network (GNN) for rich representation learning and focused fine-tuning on experimental data. In participation of the blinded ASAP-Polaris-OpenADMET antiviral ADME challenge 2025, the approach achieved competitive results, ranking fourth on aggregated mean absolute error (MAE) and tied second on aggregated Pearson R. Post-competition optimization further pushed the performance to surpass the third-place entry in MAE, without using any proprietary data or commercial featurization methods. We further explored a pretraining strategy integrating both experimental and predicted labels, showing improvements and a promising direction for pretraining on data from multiple sources. The study presents an example of new opportunities for making use of predicted labels for pretraining and applications to real-world tasks. The code and pretrained models are available on: https://github.com/LongHung-Pham/pADME.

Let G be a connected simple graph. A dominating set S⊆ V(G) is a fair dominating set in G if S=V(G) or if S≠V(G) and all vertices not in S are dominated by the same number of vertices from S, that is, |N(u)∩ S|=|N(v)∩ S|>0 for every two vertices u,v∈ V(G)∖S. A fair dominating set S of V(G) is a secure fair dominating set of G if for each u∈V(G)∖S, there exists v∈S such that uv∈E(G) and the set (S∖{v})∪ {u} is a fair dominating set of G. The minimum cardinality of a secure fair dominating set of G, denoted by γ_sfd (G), is called the secure fair domination number of G. In this paper, we initiate a study of secure fair domination in graphs and give some important results.

Let G be a simple graph. A general total coloring f of G refers to a coloring of the vertices and edges of G. Let C(x) be the set of colors of vertex x and edges incident with x under f. For a general total coloring f of G in which k colors are available, if C(u)≠C(v) for any two different vertices u and v in V(G), then f is called a k-general vertex-distinguishing total coloring of G, or a k-GVDTC of G for short. The minimum number of colors required for a GVDTC of G is denoted by χgvt(G) and is called the general vertex-distinguishing total chromatic number, or the GVDT chromatic number of G for short. GVDTCs of complete bipartite graphs are studied in this paper.

The power graph [Formula: see text] of a group [Formula: see text] is the simple and undirected graph with vertex set [Formula: see text] and with two distinct vertices being adjacent, whenever one of them is a positive power of the other in [Formula: see text]. The independence complex of a graph [Formula: see text], is the simplicial complex [Formula: see text] with the vertex set being that of [Formula: see text] and the simplices being the independent sets of [Formula: see text]. In this paper, we study the homotopy type of the independence complex of power graphs of cyclic groups of order [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct primes and [Formula: see text].

Let G=(V,E) be a finite simple graph with p=|V| vertices and q=|E| edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection f from V∪E to the consecutive integers 1,2,…,p+q, with the property that, for every vertex u∈V, one has f(u)+∑uv∈Ef(uv)=k for some magic constant k. The vertex magic total labeling is called E-super if furthermore f(E)={1,2,…,q}. A graph is called (E-super) vertex magic if it admits an (E-super) vertex magic total labeling. In this paper, we verify the existence of E-super vertex magic total labeling for a class of 3-regular graphs with a perfect matching, and we confirm the existence of such a labeling for general regular graphs of odd degree containing particular classes of 3-factors, which provides us with known and new examples. Note that Harary graphs are among the popular models used in communication networks. In 2012, G. Marimuthu and M. Balakrishnan raised a conjecture that if n>4, n≡0(mod4) and m is odd, then the Harary graph Hm,n admits an E-super vertex magic labeling. Among others, we are able to verify this conjecture except for one case while m=3 and n≡4(mod8).

Abstract Let G G be a simple graph on n n vertices with degree sequence d 1 , … , d n {d}_{1},\ldots ,{d}_{n} . Fajtlowicz ( On conjectures of Graffiti , Discrete Math. 72 (1988), 113–118) defined the temperature of a vertex v v of G G as d n − d \frac{d}{n-d} , where d d is the degree of v v . Motivated by this definition, we define the temperature index of G G , denoted by T ( G ) T\left(G) , as T ( G ) = d 1 n − d 1 + ⋯ + d n n − d n T\left(G)=\frac{{d}_{1}}{n-{d}_{1}}+\cdots +\frac{{d}_{n}}{n-{d}_{n}} . We obtain some lower bounds and upper bounds for T ( G ) T\left(G) in terms of the number of vertices, the number of edges, the maximum and the minimum vertex degree and the Zagreb index ( Z ( G ) = d 1 2 + ⋯ + d n 2 Z\left(G)={d}_{1}^{2}+\cdots +{d}_{n}^{2} ). Using these results we derive new bounds for the Zagreb index of graphs. Finally, we study the temperature index of graphs from the point of view of spectra of graphs (the eigenvalues of their adjacency matrices). In particular, we show that G G has at least one eigenvalue in the interval [ − n − δ T ( G ) − 2 m n , n − δ T ( G ) − 2 m n ] \left[-\sqrt{n-\delta }\sqrt{T\left(G)-\frac{2m}{n}},\sqrt{n-\delta }\sqrt{T\left(G)-\frac{2m}{n}}] .

Graph generalization is a powerful concept with a wide range of potential applications, while established algorithms exist for generalizing simple graphs, practical approaches for more complex graphs remain elusive. We introduce a novel formal model and algorithm (GGA) that generalizes labeled directed graphs without assuming label identity. We evaluate GGA by focusing on its information preservation relative to its input graphs, its scalability in execution, and its utility for three applications: abstract syntax trees, class graphs, and call graphs. Our findings reveal the superiority of GGA over alternative tools. GGA outperforms ASGard by an average of 5–18% on metrics related to information preservation; GGA matches 100% with diffsitter, indicating the correctness of the output. For class graphs, GGA achieves 77.1% in precision at 5, while for call graphs, it exhibits 60% in precision at 5 for a specific application problem. We also test performance for the first two applications: GGA’s execution time scales linearly with respect to the product of vertex count and edge count. Our research demonstrates the ability of GGA to preserve information in diverse applications while performing efficiently, signaling its potential to advance the field.