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Prior research has indicated that consumers' decisions are significantly influenced by online reviews. However, existing research has focused mainly on attributes (e.g., average ratings) that are not fully controlled by firms; only limited research has investigated how controllable attributes (e.g., review display formats) affect consumers. Drawing on visual perception research, the authors examine the effectiveness of two prominent graphical display formats used by major e-commerce platforms: one that displays rating distributions in a proportional format (e.g., Amazon) and one that does so in a simple format (e.g., Google). The results indicate that due to the changes in graphs' reference points caused by the shrunken x-axis in simple bar graphs, consumers respond more positively to an item when its rating distribution is displayed in a graphically simple (vs. proportional) format. This effect is moderated by the distribution's peak value (i.e., the share of the most frequent rating) and imbalance score (i.e., the difference between the share of positive and negative ratings). Furthermore, even an item's future ratings are influenced by the graphical format in which its prior ratings are displayed. The contributions to the marketing literature are discussed, and insights that can aid managers in making more informed decisions are provided. [ABSTRACT FROM AUTHOR]

The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted I (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. We show that not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a seed graph G such that H ≅ I (G). Examples of such graphs include K4 − e and K2,3. We build a series of tools to show that known i-graphs can be used to construct new i-graphs and apply these results to build other classes of i-graphs, such as block graphs, hypercubes, forests, cacti, and unicyclic graphs. [ABSTRACT FROM AUTHOR]

For a graph G, we define a total k-labeling ϕ as a combination of an edge labeling ϕe : E(G) → {1, 2, . . ., ke} and a vertex labeling ϕv : V (G) → {0, 2, . . ., 2kv}, where k = max {ke, 2kv}. The total k-labeling ϕ is called a vertex irregular reflexive k-labeling of G if any pair of vertices u, u' have distinct vertex weights wtϕ(u) ≠ wtϕ(u'), where wtϕ(u) = ϕ(u) + Σuu'∈E(G) ϕ(uu' ) for any vertex u ∈ V (G). The smallest value of k for which such a labeling exists is called the reflexive vertex strength of G, denoted by rvs(G). In this paper, we present a new lower bound for the reflexive vertex strength of any graph. We investigate the exact values of the reflexive vertex strength of generalized friendship graphs, corona product of two paths, and corona product of a cycle with isolated vertices by referring to the lower bound. This study discovers some interesting open problems that are worth further exploration. [ABSTRACT FROM AUTHOR]

We prove a version of the strong Taylor's conjecture for stable graphs: if G is a stable graph whose chromatic number is strictly greater than Æ2.@0/then G contains all finite subgraphs of Shn.!/and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model-theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability. [ABSTRACT FROM AUTHOR]

Root cause analysis (RCA) plays an essential role in quality problem solving (QPS). Due to the difficulty of obtaining causal knowledge of quality problems, companies often rely on expert experience and conventional RCA tools when conducting RCA. Rich QPS data have remained mostly untapped but provide the potential for causal knowledge mining, while the semistructured nature of these data poses enormous challenges to this task. Thus, we propose a data-driven framework to mine large-scale causalities between quality problems and production factors from QPS data and exploit a causal knowledge graph for quality problems (QPCKG) to express these causalities. We first classify QPS data to identify the data containing causality. The causal linguistic patterns are then employed to extract cause slots and effect slots from these data. Subsequently, we apply the BiLSTM-CRF to extract the core content of problems. A vertex fusion method is last proposed to integrate discrete causalities into QPCKG. The approach is validated in a real-world application at a leading automotive company. Three potential applications of the QPCKG are demonstrated for quality diagnosis and prediction. The QPCKG reveals a grand picture of the core interaction mechanism of product quality and production factors and provides decision-making support for RCA. [ABSTRACT FROM AUTHOR]

We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high enough cardinality will have a single model in any high enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation. [ABSTRACT FROM AUTHOR]

Online video platforms such as YouTube feature visual cues on progress bars to spotlight standout segments of videos. These cues, designed to steer users toward intriguing content moments, raise questions about their overarching impact on video consumption patterns. In this study, we delve into inconsumption information cues (ICICs), indicators that depict fluctuating video quality in real time. Drawing from a natural experiment on a live streaming platform, we evaluate the effects of engagement graphs. These are visual timelines that emphasize moments favored by prior viewers in video-on-demand (VOD) content, which represent full replays of live stream sessions. Notably, these graphs are only accessible to iOS users, leaving Android users without access. Our results show that ICICs enhance the viewership of VODs and live streams. Moreover, viewers tend to spend more time immersed in live broadcasts, suggesting heightened content appeal. Yet, these engagement graphs do not prompt users to donate more virtual gifts, a vital income stream for streamers. Even with the introduction of ICICs, while there is a rise in video production, the inherent structure of live streams remained largely unchanged. We conclude by discussing the academic and managerial implications of these findings. [ABSTRACT FROM AUTHOR]

In this paper, we discuss spectrum and energy of Arithmetic function signed graphs with respect to an arithmetical function and present some results. [ABSTRACT FROM AUTHOR]

This study aims at the investigation of student's problems in understandings diagrams and graphs in the high schools of district Mardan. Objectives of the study were to assess students' attitudes towards diagram and graphs, to find difficulties in understandings diagrams and graphs, to find the nature of difficulties in students physics work and to investigate reasons about complexity of graph and diagrams. Research design is qualitative in nature. Qualitative method was used to understand participant experience with error analysis. The study was conducted in district Mardan. Random sampling techniques were used to collect qualitative data. Participant of the study was 240 students of grade 9th from four selected schools in which two schools were selected from public sector and two from private sector. 60 students were selected from each school. Content analysis was used to identify student's error. Data was collected through interview or test from 40 students of Grade 9th. Finding was obtained after analysis of data; conclusion was drawn and recommendations were made accordingly. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This article argues that a reliance on material texts tied to the concept of improvement—such as picturesque engravings, diagrams, and account books—pushed the early abolitionist movement toward a reforming, ameliorationist ethic that disavowed revolutionary action and immediate emancipation. Although the term improvement had broad social applicability by the late eighteenth century, its original connections to land management made it an especially important concept for the abolitionist debate. Integrating book history with studies of enslavement and abolition, I show how abolitionists' use of visual-textual forms such as diagrams and account books created an emphasis on gradual improvement. In response, Black abolitionists both emphasized concepts other than improvement and presented their works in different material forms. Olaudah Equiano and Quobna Ottabah Cugoano adapted the genre of autobiography to demonstrate how their own ostensible improvements did not have the effects anticipated by the logic of white abolitionists, ultimately undercutting the usefulness of improvement as a guiding concept for abolitionism. [ABSTRACT FROM AUTHOR]

We consider the problem of counting the number of copies of a fixed graph H within an input graph G. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input G has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g., planar graphs), as well as graphs generated by various random processes (e.g., preferential attachment graphs). We say that H is easy if there is a linear-time algorithm for counting the number of copies of H in an input G of bounded degeneracy. A seminal result of Chiba and Nishizeki from '85 states that every H on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all H on 5 vertices and further proved that for every k > 5 there is a k-vertex H which is not easy. They left open the natural problem of characterizing all easy graphs H. Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph H to be easy. Here, we show that this sufficient condition is also necessary, thus fully answering the Bera-Pashanasangi-Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms. [ABSTRACT FROM AUTHOR]

We consider the evaluation of first-order queries over classes of databases that are nowhere dense. The notion of nowhere dense classes was introduced by Nešetřil and Ossona de Mendez as a formalization of classes of "sparse" graphs and generalizes many well-known classes of graphs, such as classes of bounded degree, bounded tree-width, or bounded expansion. It has recently been shown by Grohe, Kreutzer, and Siebertz that over nowhere dense classes of databases, first-order sentences can be evaluated in pseudo-linear time (pseudo-linear time means that for all - there exists an algorithm working in time O(n1+ε), where n is the size of the database). For first-order queries of higher arities, we show that over any nowhere dense class of databases, the set of their solutions can be enumerated with constant delay after a pseudo-linear time preprocessing. In the same context, we also show that after a pseudo-linear time preprocessing we can, on input of a tuple, test in constant time whether it is a solution to the query. [ABSTRACT FROM AUTHOR]

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA'14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, Kt -free unit d-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d-contractions, witness that the twin-width is at most d. We show that FO model checking, that is deciding if a given first-order formula ϕ evaluates to true for a given binary structure G on a domain D, is FPT in |ϕ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d-contraction sequence for G, our algorithm runs in time f (d, |ϕ|) · |D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved under FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS'15]. [ABSTRACT FROM AUTHOR]

In the k-cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O(n2k) time. However, lower bounds from conjectures about the k-clique problem imply that O(n(1-o(1))k) time is likely needed. Recent results of Gupta, Lee, and Li have given new algorithms for general k-cut in n1.98k+O(1) time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In thiswork, we resolve the problem for general graphs. We showthat the Contraction Algorithm of Karger outputs any fixed k-cut of weight a-k with probability Ok (n -ak), where -k denotes the minimum k-cut weight. This also gives an extremal bound of Ok (nk) on the number of minimum k-cuts and an algorithm to compute -k with roughly nk polylog(n) runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k-clique. The firstmain ingredient in our result is an extremal bound on the number of cuts of weight less than 2-k/k, using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks--and how the average degree evolves--in the Karger process. [ABSTRACT FROM AUTHOR]

The Hosoya index counts the number of independent edge sets in a graph, that is the number of subsets of the edge set such that no two edges in the subset share a vertex. Moreover, the Hosoya index gives important details on a graph's structural properties, including its connectivity. It has applications in a variety of fields, including computational biology, networking, and chemistry. In our article, we study Hosoya indiex of amalgamation of cycles and edge-amalgamation of cycles. Moreover, in this article we study the restricted Hosoya polynomial of amalgamation of cycles and we also give the general form of topological index. [ABSTRACT FROM AUTHOR]

We describe various diagram algebras and their representation theory using cellular algebras of Graham and Lehrer and the decomposition into half diagrams. In particular, we show the diagram algebras surveyed here are all cellular algebras and parameterize their cell modules. We give a new construction to build new cellular algebras from a general cellular algebra and subalgebras of the rook Brauer algebra that we call the cellular wreath product. [ABSTRACT FROM AUTHOR]

There are various circumstances where it is important to simultaneously monitor or control two or more related quality characteristics. Independently tracking these quality characteristics might be quite deceptive. Hotelling's T2 chart, in which the T2 statistics are generated using the classical estimates of location and scatter, is the most well-known multivariate process monitoring and control approach. It is well known that the existence of outliers in a dataset has a significant impact on classical estimators. Any statistic that is computed using the classical estimates will be distorted by even a single outlier. The non-robustness issue is investigated in this study, which also suggests four robust bivariate control charts based on the robust Gnandesikan-Kettenring estimator. This study employs four highly robust scale estimators, with the best breakdown point, namely the Qn estimator, Sn estimator, MAD estimator, and τ estimator, in order to robustify the Gnandesikan- Kettenring estimator. Through the use of a Monte Carlo simulation and a real-life data, the performance of the suggested control charts is assessed. The four techniques all outperform the traditional method and provide greater computing efficiency. [ABSTRACT FROM AUTHOR]

When applied to Tai varieties beyond Southwestern Tai, Gedney's tone box framework and its revised versions have several shortcomings, the most notable of which is the lack of consideration of diachronic issues that give rise to more complicated tonal correspondences. This paper proposes a more widely applicable Tai tone box chart with one additional tier based on the more probable tone-conditioning phonetic nature of onsets found in non-Southwestern Tai varieties. This Tai tone box chart has been designed to be paired with an ancillary chart to explain the essential diachronic processes of different Tai varieties. It thus provides insights into the revision of Proto-Tai phonological reconstructions based on tonology and support for the early-stage Southern-Northern Tai division and the Southwestern-Central Tai versus Yongnan Zhuang-Saek-Northern Zhuang divisions. [ABSTRACT FROM AUTHOR]

Much research has reported on students' difficulties with solving related rates problems in calculus. In an effort to generate a resource that could potentially address some of these difficulties from a teaching standpoint, a questionnaire about effective instructional approaches related to the teaching of related rates problems, among other things, was administered to 14 veteran calculus instructors. Analysis of the responses provided by the instructors revealed that all the instructors considered the use of diagrams to be helpful when solving related rates problems. Furthermore, a majority of these instructors noted that introducing a set of steps (i.e., a guideline), during classroom instruction, that students could follow when solving related rates problems is helpful for students when working with this type of problems. These instructors further identified strengths and weaknesses in the way related rates problems are typically presented in calculus textbooks. Implications for instruction are included. [ABSTRACT FROM AUTHOR]

The natural language to SQL (NL2SQL) task is an emerging research area that aims to transform a natural language with a given database into an SQL query. The earlier approaches were to process the input into a heterogeneous graph. However, previous models failed to distinguish the types of multi-hop connections of the heterogeneous graph, which tended to ignore crucial semantic path information. To this end, a two-layer attention network is presented to focus on essential neighbor nodes and mine enlightening semantic paths for feature encoding. The weighted edge is introduced for schema linking to connect the nodes with semantic similarity. In the decoding phase, a rule-based pruning strategy is offered to refine the generated SQL queries. From the experimental results, the approach is shown to learn a good encoding representation and decode the representation to generate results with practical meaning. [ABSTRACT FROM AUTHOR]

The energy of a graph G, denoted by E(G), is defined as the sum of absolute values of all eigenvalues of G. A graph G is called reciprocal if 1/≥ is an eigenvalue of G whenever λ is an eigenvalue of G. Further, if λ and 1 λ have the same multiplicities, for each eigenvalue λ, then it is called strong reciprocal. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631{633), it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree ∞(G) whose adjacency matrix is non-singular, E(G) λ Δ(G) + ∞(G) and the equality holds if and only if G is a complete graph. Here, we prove the validity of this conjecture for some strong reciprocal graphs. Moreover, we show that if G is a strong reciprocal graph, then E(G) λ Δ(G) + ∞(G) - 1 2. Recently, it has been proved that if G is a reciprocal graph of order n and its spectral radius, ?, is at least 4λmin, where λmin is the smallest absolute value of eigenvalues of G, then E(G) λ n + 1 2. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption. [ABSTRACT FROM AUTHOR]

The article discusses the creation, use, disuse, and disappearance of the Institutional and Cultural Supports for Battering Chart, which was designed as a close companion of the Power and Control Wheel, an iconic image in the anti-gender violence field. Topics include the ubiquity and influence of the Wheel and the obscurity of the Chart, history of the Wheel and Chart, and factors that likely contributed to the Chart's disappearance.

Summary: We present visual proofs illustrating certain identities relating hexagonal numbers to other figurate numbers. [ABSTRACT FROM AUTHOR]

A derivative structure is a nonequivalent substitutional atomic configuration derived from a given primitive cell. The enumeration of derivative structures plays an essential role in searching for the ground states in multicomponent systems. However, it is computationally difficult to enumerate derivative structures if the number of derivative structures of a target system becomes huge. In this study, we introduce a novel compact data structure of the zero-suppressed binary decision diagram (ZDD) for enumerating derivative structures much more efficiently. We show its simple applications to the enumeration of structures derived from the face-centered cubic and hexagonal close-packed lattices in binary, ternary, and quaternary systems. The present ZDD-based procedure should contribute to computational approaches based on derivative structures in physics and materials science. [ABSTRACT FROM AUTHOR]

Analysis of first-order reversal curves (FORCs) is a powerful tool to probe irreversible switching events in nanomagnet assemblies. As in essence switching events are related to the intrinsic properties of the constituents and their interactions, the resulting FORC diagrams contain much information that can be cross-linked and complex to deconvolute. In order to quantify the relevant parameters that drive the FORC diagrams of arrays of perpendicularly magnetized nanomagnets, we present step-by-step simulations of assemblies of hysterons to determine the specific signatures related to different known inputs. While we explored the consequences of dipolar interactions using either mean field or magnetostatic approaches, we completed by taking the hysteron switching field distribution (SFD) as either normal or lognormal. We demonstrated that the transition between FORC diagrams composed of an isolated interaction field distribution (IFD) and a wishbone shape operates via the SFD deviation, σ H s w , in the presence of a weakly dispersed interaction field. In the presence of a magnetostatic interaction field, the IFD profile is peaked and a coercive field distribution (CFD) sums to the IFD as σ H s w increases. A transition between IFD + CFD and wishbone shapes is clearly demonstrated as a function of the interaction field deviation σ H i n t . In addition, we demonstrate that whatever the considered cases, σ H s w can be quantitatively extracted from the FORC diagrams within an error inferior to 10%. These findings are of interest for dipolar coupled perpendicularly magnetized nanomagnets, as in assemblies of magnetic nanowires and nanopillars, as well as bit patterned media. [ABSTRACT FROM AUTHOR]

On the basis of the screened 29 hybrid halide compounds from our prior study [Y. Li and K. Yang, Energy Environ. Sci. 12, 2233–2243 (2019)], here, we reported a systematic computational study of the stability diagram, defect tolerance, and optical absorption coefficients for these candidate materials using high-throughput first-principles calculations. We took two exemplary compounds, MA2SnI4 and MA3Sb2I9, as examples to show the computational process, and they are discussed in detail. This work is expected to provide a detailed guide for further experimental synthesis and characterization, with the potential to develop novel lead-free optoelectronic devices. [ABSTRACT FROM AUTHOR]

The generalized dynamic Tullock contest model with two homogeneous participants is established, in which both players have the same valuation of winning rewards and losing rewards. Firstly, the unique symmetric equilibrium point of the system is obtained by calculation and its local stability condition is given based on the Jury criterion. Then, two paths of the system from stability to chaos, namely flip bifurcation and Neimark-Sacker bifurcation, are analyzed by using the two-dimensional parametric bifurcation diagram. Meanwhile, the abundant Arnold tongues in the two-dimensional parametric bifurcation diagram are analyzed. Finally, the phenomenon of multistability of the system is illustrated through the basin of attraction, and the contact bifurcation occurs during the evolution of the basin of attraction with varying parameters. [ABSTRACT FROM AUTHOR]

Based on DBR, a pressure vessel and standard vessel are designed using a public view and diagrams. They are formulated to a standard specification using the ASME pressure vessel code rules and the European standard EN13445. The pressure is determined under influence of a closed end. There are limitations to the structure of the container. Using DBA, several analyses were performed based on FEM to provide parameters to complete the DOE, and combined with DFA and DFM to verify that the analysis was conducive to giving users a good analysis plan. The core premise of this study is to simplify operation, to provide an exemplary user interface, observe various parameters according to requirements, meet pressure requirements during the reaction of polymer composite materials, and ensure that the temperature, strength, and manufacturing cost are within safety factors, to provide a good user experience. [ABSTRACT FROM AUTHOR]

A simple theoretical model of network evolution is discussed here. In each step, we add a new vertex to the graph and it is allowed to connect it to maximum degree vertices (hubs) only. Given a constant p, the probability of such a connection is p for any hub. The initial (non-random) graph G1 is arbitrary but here we investigate mostly the case when G1 has one vertex. We solve here some particular cases of the problem, using enumeration methods. We obtain not limit theorems but exact results for the parameters discussed. [ABSTRACT FROM AUTHOR]

Cop Robber game is a two player game played on an undirected graph. In this game, the cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. We study cop-edge critical graphs, i.e. graphs G such that for any edge e in E(G) either c(G -e) < c(G) or c(G-e) > c(G). In this article, we study the edge criticality of generalized Petersen graphs and Paley graphs. [ABSTRACT FROM AUTHOR]

Let D be a simple connected digraph with n vertices and m arcs and let W (D) = (D, w) be the weighted digraph corresponding to D, where the weights are taken from the set of non-zero real numbers. Let v1,V2,...,Vn be the eigenvalues of the skew Laplacian weighted matrix SLW (D) of the weighted digraph W(D). In this paper, we discuss the skew Laplacian energy SLEW(D) of weighted digraphs and obtain the skew Laplacian energy of the weighted star W(K1,n) for some fixed orientation to the weighted arcs. We obtain lower and upper bounds for SLEW(D) and show the existence of weighted digraphs attaining these bounds. [ABSTRACT FROM AUTHOR]

Let S=(G,σ) be a signed graph. A function f:V→{0,1,2} is a Roman dominating function on S if (i) for each v € V, f(N[v])=f(v)+∑u€N(v)σ(uv)f(u)≥1 and (ii) for each vertex v with f(v)=0 there exists a vertex u €N+(v) such that f(u)=2. In this paper we initiate a study on Roman dominating function on signed graphs. We characterise the signed paths, cycles and stars that admit a Roman dominating function. [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a simple graph with vertex set V and edge set E. An {outer-independent total 2 -rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1,2} such that the following conditions hold: (i) for any vertex v with f(v)=∅ we have ⋃u∈NG(v)f(u)={1,2}, (ii) the set of all vertices v∈V(G) with f(v)=∅ is independent and (iii) {v∣f(v)≠∅} has no isolated vertex. The outer-independent total 2 -rainbow domination number of G, denoted by γoitr2(G), is the minimum value of ω(f)=∑v∈V(G)|f(v)| over all such functions f. In this paper, we study the outer-independent total 2 -rainbow domination number of G and classify all graphs with outer-independent total 2 -ainbow domination number belonging to the set {2,3,n}. Among other results, we present some sharp bounds concerning the invariant. [ABSTRACT FROM AUTHOR]

A coalition in a graph G=(V,E) consists of two disjoint sets V1 and V2 of vertices, such that neither V1 nor V2 is a dominating set, but the union V1∪V2 is a dominating set of G. A coalition partition in a graph G of order n=|V| is a vertex partition π=V1,V2,...,Vk such that every set Vi either is a dominating set consisting of a single vertex of degree n-1, or is not a dominating set but forms a coalition with another set Vj. Associated with every coalition partition π of a graph G is a graph called the coalition graph of G with respect to π, denoted CG(G,π), the vertices of which correspond one-to-one with the sets V1,V2,...,Vk of π and two vertices are adjacent in CG(G,π) if and only if their corresponding sets in π form a coalition. In this paper, we initiate the study of coalition graphs and we show that every graph is a coalition graph. [ABSTRACT FROM AUTHOR]

The 2S 3 transformation, which was first described for positive integers, has been defined for dyadic rational numbers in the op en interval (0, 1) in this study. The set of dyadic rational numbers is a Prüfer 2-group. For the dyadic 2S3 transformation Tds(x), the restricted multiplicative and additive properties have been established. Graph parameters are used to generate more combinatorial outcomes for these properties. The relationship between the SM dyadic sum graph's automorphism group and the symmetric group has been investigated. [ABSTRACT FROM AUTHOR]

The concept of topology defined on a set can be extended to the field of graph theory by defining the notion of graph topologies on graphs where we consider a collection of subgraphs of a graph G in such a way that this collection satisfies the three conditions stated similarly to that of the three axioms of point-set topology. This paper discusses an introduction and basic concepts to the graph topology. A subgraph of G is said to be open if it is in the graph topology TG. The paper also introduces the concept of the closed graph and the closure of graph topology in graph topological space using the ideas of decomposition-complement and neighborhood-complement. [ABSTRACT FROM AUTHOR]

For a (molecular) graph, the second Zagreb index M2(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices. Roman dominating function RDF of G is a function f:V(G)→{0,1,2} satisfying the condition that every vertex with label 0 is adjacent to a vertex with label 2. The weight of an RDF f is w(f)=∑v∈V(G)f(v). The Roman domination number of G, denoted by γR(G), is the minimum weight among all RDF in G. In this paper, we present a lower bound on the second Zagreb index of trees with n vertices and Roman domination number and thus settle one problem given in [On the Zagreb indices of graphs with given Roman domination number, Commun. Comb. Optim. DOI: 10.22049/CCO.2021.27439.1263 (article in press)]. [ABSTRACT FROM AUTHOR]

In this paper, we have punctured unit Zq -Simplex code and constructed a new code called unit Zq -Simplex code of type α. In particular, we find the parameters of these codes and have proved that it is an [ϕ(q)+2, 2, ϕ(q)+2-ϕ(q)ϕ(p)] Zq -linear code if k=2 and [ϕ(q)k-1ϕ(q)-1+ϕ(q)k-2, k, ϕ(q)k-1ϕ(q)-1+ϕ(q)k-2-(ϕ(q)ϕ(p))(ϕ(q)k-1-1ϕ(q)-1+ϕ(q)k-3)] Zq -linear code if k≥3, where p is the smallest prime divisor of q. For q is a prime power and rank k=3, we have given the weight distribution of unit Zq -Simplex codes of type α. Also, we have introduced some new code from Zq -Simplex code called zero divisor Zq -Simplex code and proved that it is an [ρk-1ρ-1,k,ρk-1ρ-1-(ρ(k-1)-1ρ-1)(qp)] Zq -linear code, where ρ=q-ϕ(q) and p is the smallest prime divisor of q. Further, we obtain weight distribution of zero divisor Zq -Simplex code for rank k=3 and q is a prime power. [ABSTRACT FROM AUTHOR]

In this paper, we have characterized the commutative rings with unity for which line signed graph of a signed unit graph is balanced and consistent. To do this, we first establish some sufficient conditions for balance and consistency of line signed graph of signed unit graphs. [ABSTRACT FROM AUTHOR]

The Sombor index of the graph G is a degree based topological index, defined as SO=∑uv∈E(G)d²u+d²v--√, where du is the degree of the vertex u, and E(G) is the edge set of G. Bounds on SO are established in terms of graph energy, size of minimum vertex cover, matching number, and induced matching number. [ABSTRACT FROM AUTHOR]

Copyright of Duzce University Journal of Science & Technology is the property of Duzce University Journal of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-4 internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations, we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space M0;4. We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations. [ABSTRACT FROM AUTHOR]