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Moran's I is a spatial autocorrelation measure of univariate spatial data. Therefore, even if p spatial data exist, we can only obtain p values for Moran's I. In other words, Moran's I cannot measure the degree of spatial autocorrelation of multivariate spatial data as a single value. This paper addresses this issue. That is, we extend Moran's I so that it can measure the degree of spatial autocorrelation of multivariate spatial data as a single value. In addition, since the local version of Moran's I has the same problem, we extend it as well. Then, we establish their properties, which are fundamental for applied work. Numerical illustrations of the theoretical results obtained in the paper are also provided. [ABSTRACT FROM AUTHOR]

Consider a finite directed graph without cycles in which the arrows are weighted by positive weights. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context of fraud detection. The idea is that the new distance explicitly takes into account the size of the paths in the calculations. It has the distinct characteristic that, when calculated along the same path, it may result in a shorter distance between far-apart vertices than between adjacent ones. This property can be particularly useful for modeling scenarios where the connections between vertices are obscured by numerous intermediate vertices, such as in cases of financial fraud. For example, to hide dirty money from financial authorities, fraudsters often use multiple institutions, banks, and intermediaries between the source of the money and its final recipient. Our distance would serve to make such situations explicit. Thus, although our algorithm is based on arguments similar to those at work for the Bellman–Ford and Dijkstra methods, it is in fact essentially different, since the calculation formula contains a weight that explicitly depends on the number of intermediate vertices. This fact totally conditions the algorithm, because longer paths could provide shorter distances—contrary to the classical algorithms mentioned above. We lay out the appropriate framework for its computation, showing the constraints and requirements for its use, along with some illustrative examples. [ABSTRACT FROM AUTHOR]

A mathematical procedure enabling the transformation of an arbitrary tessellation of a surface into a bi-colored, complete graph is introduced. Polygons constituting the tessellation are represented by vertices of the graphs. Vertices of the graphs are connected by two kinds of links/edges, namely, by a green link, when polygons have the same number of sides, and by a red link, when the polygons have a different number of sides. This procedure gives rise to a semi-transitive, complete, bi-colored Ramsey graph. The Ramsey semi-transitive number was established as R t r a n s (3 , 3) = 5 Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits ζ = lim N → ∞ N g N r , where N is the total number of green and red seeds, N g and N r , were found ζ = 0.272 ± 0.001 (Voronoi) and ζ = 0.47 ± 0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as S = 1.690 ± 0.001 and for the Poisson line tessellation as S = 1.265 ± 0.015. The main contribution of the paper is the calculation of the Shannon entropy of the random point process and the establishment of the new bi-colored Ramsey graph on top of the tessellations. [ABSTRACT FROM AUTHOR]

In this paper, we initiate a concept of graph-proximal functions. Furthermore, we give a notion of being generalized Geraghty dominating for a pair of mappings. This permits us to establish the existence of and unique results for a common best proximity point of complete metric space. Additionally, we give a concrete example and corollaries related to the main theorem. In particular, we apply our main results to the case of metric spaces equipped with a reflexive binary relation. Finally, we demonstrate the existence of a solution to boundary value problems of particular second-order differential equations. [ABSTRACT FROM AUTHOR]

We introduce a unifying approach for invariants of finite matroids that count mappings to a finite set. The aim of this paper is to show that if the cardinalities of mappings with fixed values on a restricted set satisfy contraction–deletion rules, then there is a relation among them that can be expressed in terms of linear algebra. In this way, we study regular chain groups, nowhere-zero flows and tensions on graphs, and acyclic and totally cyclic orientations of oriented matroids and graphs. [ABSTRACT FROM AUTHOR]

Quantum codes are crucial building blocks of quantum computers. With a self-dual quantum code is attached, canonically, a unique stabilised quantum state. Improving on a previous publication, we show how to determine the coefficients on the basis of kets in these states. Two important ingredients of the proof are algebraic graph theory and quadratic forms. The Arf invariant, in particular, plays a significant role. [ABSTRACT FROM AUTHOR]

Let Heff (Veff, Eeff) be a finite simple connected graph of order m with vertex set Veff and edge set Eeff. A dominating set S d s ⊆ V e f f is called an efficiently dominating set if, for every vertex u a ∈ V G ,   N G u a ∩ S d s = 1 —where NG [ua] denotes the closed neighborhood of the vertex ua. Using efficient domination techniques and labelling, we constructed the fuzzy network. An algorithm has been framed to encrypt and decrypt the secret information present in the network, and furthermore, the algorithm has been given in pseudocode. The mathematical modelling of a strong fuzzy network is defined and constructed to elude the burgeoning intruder. Using the study of the efficient domination of fuzzy graphs, this domination parameter plays a nuanced role in encrypting and decrypting the framed network. One of the main purposes of fuzzy networks is encryption, so one of our contributions to this research is to build a novel combinatorial technique to encrypt and decode the built-in fuzzy network with a secret number utilizing effective domination. An illustration with an appropriate secret message is provided along with the encryption and decryption algorithms. Furthermore, we continued this study in intuitionistic fuzzy networks. [ABSTRACT FROM AUTHOR]

Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph Ω provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first Zagreb index of a graph Ω is defined as the product of the sum of the α th powers of the vertex degrees of Ω , where α is a real number such that α ≠ 0 and α ≠ 1 . The focus of this work is on the extremal graphs for several classes of graphs including trees, unicyclic, and bicyclic graphs, with respect to the generalized multiplicative first Zagreb index. In the initial step, we identify a set of operations that either increases or decreases the generalized multiplicative first Zagreb index for graphs. We then involve analysis of the generalized multiplicative first Zagreb index achieving sharp bounds by characterizing the maximum or minimum graphs for those classes. We present applications of the generalized multiplicative first Zagreb index Π 1 α for predicting the π -electronic energy E π (β) of benzenoid hydrocarbons. In particular, we answer the question concerning the value of α for which the predictive potential of Π 1 α with E π for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that Π 1 α correlates with E π of lower benzenoid hydrocarbons with correlation coefficient ρ = − 0.998 , if α = − 0.00496 . In QSPR modeling, the value ρ = − 0.998 is considered to be considerably significant. [ABSTRACT FROM AUTHOR]

A banknote is a currency issued by a country, and it was first introduced in the 16th century. The counterfeiting of banknotes by cunning criminals became a great challenge with the current advanced technology. Forensic scientists are using chemical methods, such as Fourier transform infrared (FTIR) spectroscopy for differentiating genuine and counterfeit banknotes. However, the FTIR spectra of banknotes may require further pattern recognition analysis due to their high similarities. In this paper, a fuzzy graph-based algorithm for authentication of the FTIR spectrum, namely chemometrics fuzzy autocatalytic set (c-FACS), is used to discriminate between genuine and counterfeit hundred Ringgit Malaysian (RM100) banknotes. The results show that the genuine and counterfeit RM100 banknotes have slightly distinct patterns when analyzed using c-FACS. In addition, the results are compared with RM50 banknotes, and the results reveal that the nodes or dominant axis varies between the two banknotes. To verify the reliability of the results, the results obtained via c-FACS are compared with principal component analysis (PCA). The c-FACS showed better performances as compared to PCA in terms of time consumption and observation. Thus, the c-FACS has the ability to assist forensic investigations involving banknote counterfeiting crimes. [ABSTRACT FROM AUTHOR]

The game of pursuit and evasion, when played on graphs, is often referred to as the game of cops and robbers. This classical version of the game has been completely solved by Nowakowski and Winkler, who gave the exact class of graphs for which the pursuer can win the game (cop-win). When the graph does not satisfy the dismantlability property, Nowakowski and Winkler's Theorem does not apply. In this paper, we give some weaker notions of cop-win and robber-win graphs. In particular, we fix the number of cops to be equal to one, and we ask whether there exist sets of initial conditions for which the graph can be cop-win or robber-win. We propose some open questions related to this initial condition problem with the goal of developing both structural characterizations and algorithms that are computable in polynomial time (P) and that can solve weakly cop-win and weakly- robber-win graphs. [ABSTRACT FROM AUTHOR]

We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subset and prove uniqueness theorems from paired-dense nodal subsets in interior subintervals having a central vertex. In particular, we obtain some uniqueness theorems by replacing the information of nodal data on some fixed edge with part of the eigenvalues under some conditions. [ABSTRACT FROM AUTHOR]

For a connected graph G on n vertices, recall that the reciprocal distance signless Laplacian matrix of G is defined to be R Q (G) = R T (G) + R D (G) , where R D (G) is the reciprocal distance matrix, R T (G) = d i a g (R T 1 , R T 2 , ⋯ , R T n) and R T i is the reciprocal distance degree of vertex v i . In 2022, generalized reciprocal distance matrix, which is defined by R D α (G) = α R T (G) + (1 − α) R D (G) , α ∈ [ 0 , 1 ] , was introduced. In this paper, we give some bounds on the spectral radius of R D α (G) and characterize its extremal graph. In addition, we also give the generalized reciprocal distance spectral radius of line graph L (G) . [ABSTRACT FROM AUTHOR]

For a simple connected graph G = (V , E) , an ordered set W ⊆ V , is called a resolving set of G if for every pair of two distinct vertices u and v, there is an element w in W such that d (u , w) ≠ d (v , w) . A metric basis of G is a resolving set of G with minimum cardinality. The metric dimension of G is the cardinality of a metric basis and it is denoted by β (G) . In this article, we determine the metric dimension of power of finite paths and characterize all metric bases for the same. [ABSTRACT FROM AUTHOR]

Consider the following problem. In a school with three classes containing n students each, given that their genders are unknown, find the minimum possible number of triples of same-gender students—not all of which are from the same class. Muaengwaeng asked this question and conjectured that the minimum scenario occurs when the classes are all boys, all girls and half-and-half. In this paper, we solve many generalizations of the problem including when the school has more than three classes, when triples are replaced by groups of larger sizes, when the classes are of different sizes, and when gender is replaced by other non-binary attributes. [ABSTRACT FROM AUTHOR]

In the paper, the crossing number of the join product G * + D n for the disconnected graph G * consisting of two components isomorphic to K 2 and K 3 is given, where D n consists of n isolated vertices. Presented proofs are completed with the help of the graph of configurations that is a graphical representation of minimum numbers of crossings between two different subgraphs whose edges do not cross the edges of G * . For the first time, multiple symmetry between configurations are presented as parity properties. We also determine crossing numbers of join products of G * with paths P n and cycles C n on n vertices by adding new edges joining vertices of D n . [ABSTRACT FROM AUTHOR]

Graphs of order n with fault-tolerant metric dimension n have recently been characterized.This paper points out an error in the proof of this characterization. We show that the complete multipartite graphs also have the fault-tolerant metric dimension n, which provides an infinite family of counterexamples to the characterization. Furthermore, we find exact values of the metric, edge metric, mixed-metric dimensions, the domination number, locating-dominating number, and metric-locating-dominating number for the complete multipartite graphs. These results generalize various results in the literature from complete bipartite to complete multipartite graphs. [ABSTRACT FROM AUTHOR]

We give a family of counterexamples of a theorem on a new upper bound for the α-indices of graphs in the paper that is mentioned in the title. We also provide a new upper bound for corrigendum. [ABSTRACT FROM AUTHOR]

Persistent homology enables fast and computable comparison of topological objects. We give some instances of a recent extension of the theory of persistence, guaranteeing robustness and computability for relevant data types, like simple graphs and digraphs. We focus on categorical persistence functions that allow us to study in full generality strong kinds of connectedness—clique communities, k-vertex, and k-edge connectedness—directly on simple graphs and strong connectedness in digraphs. [ABSTRACT FROM AUTHOR]

We establish fixed point, stability and genericity theorems for strict contractions on complete metric spaces with graphs. [ABSTRACT FROM AUTHOR]

This study analyzes the spillover effects of volatility in the Russian stock market. The paper applies the Diebold–Yilmaz connectedness methodology to characterize volatility spillovers between Russian assets. The spectral representation of the forecast variance decomposition proposed by Baruník and Křehlik is used to describe the connectivity in short-term (up to 5 days), medium-term (6–20 days) and long-term (more than 20 days) time frequencies. Additionally, two new augmented models are developed and applied to evaluate conditional spillover effects in different sectors of the Russian economy for the period from January 2012 to June 2021. It is shown that spillover effects increase significantly during political and economic crises and decrease during periods of relative stability. The rising of the overall level of spillovers in the Russian stock market coincides in time with the political crisis of 2014, the intensification of anti-Russian sanctions in 2018 and the fall in oil prices and the start of the pandemic in 2020. With the consideration of the augmented models it can be argued that a significant part of the long-term spillover effects on the Russian stock market may be caused by the influence of external economic and political factors. However, volatility spillovers generated by internal Russian idiosyncratic shocks are short-term. Thus, the proposed approach provides new information on the impact of external factors on volatility spillovers in the Russian stock market. [ABSTRACT FROM AUTHOR]

We consider a general second order self-adjoint elliptic operator on an arbitrary metric graph, to which a small graph is glued. This small graph is obtained via rescaling a given fixed graph γ by a small positive parameter ε. The coefficients in the differential expression are varying, and they, as well as the matrices in the boundary conditions, can also depend on ε and we assume that this dependence is analytic. We introduce a special operator on a certain extension of the graph γ and assume that this operator has no embedded eigenvalues at the threshold of its essential spectrum. It is known that under such assumption the perturbed operator converges to a certain limiting operator. Our main results establish the convergence of the spectrum of the perturbed operator to that of the limiting operator. The convergence of the spectral projectors is proved as well. We show that the eigenvalues of the perturbed operator converging to limiting discrete eigenvalues are analytic in ε and the same is true for the associated perturbed eigenfunctions. We provide an effective recurrent algorithm for determining all coefficients in the Taylor series for the perturbed eigenvalues and eigenfunctions. [ABSTRACT FROM AUTHOR]

Often for understanding a structure, other closely related structures with the former are associated. An example of this is the study of hyperspaces. In this paper, we give necessary and sufficient conditions for the existence of finitely-dimensional maximal free cells in the hyperspace C (G) of a dendrite G; then, we give necessary and sufficient conditions so that the aforementioned result can be applied when G is a dendroid. Furthermore, we prove that the arc is the unique arcwise connected, compact, and metric space X for which the anchored hyperspace C p (X) is an arc for some p ∈ X . [ABSTRACT FROM AUTHOR]

The paper presents an analysis and overview of vertex–frequency analysis, an emerging area in graph signal processing. A strong formal link of this area to classical time–frequency analysis is provided. Vertex–frequency localization-based approaches to analyzing signals on the graph emerged as a response to challenges of analysis of big data on irregular domains. Graph signals are either localized in the vertex domain before the spectral analysis is performed or are localized in the spectral domain prior to the inverse graph Fourier transform is applied. The latter approach is the spectral form of the vertex–frequency analysis, and it will be considered in this paper since the spectral domain for signal localization is well ordered and thus simpler for application to the graph signals. The localized graph Fourier transform is defined based on its counterpart, the short-time Fourier transform, in classical signal analysis. We consider various spectral window forms based on which these transforms can tackle the localized signal behavior. Conditions for the signal reconstruction, known as the overlap-and-add (OLA) and weighted overlap-and-add (WOLA) methods, are also considered. Since the graphs can be very large, the realizations of vertex–frequency representations using polynomial form localization have a particular significance. These forms use only very localized vertex domains, and do not require either the graph Fourier transform or the inverse graph Fourier transform, are computationally efficient. These kinds of implementations are then applied to classical time–frequency analysis since their simplicity can be very attractive for the implementation in the case of large time-domain signals. Spectral varying forms of the localization functions are presented as well. These spectral varying forms are related to the wavelet transform. For completeness, the inversion and signal reconstruction are discussed as well. The presented theory is illustrated and demonstrated on numerical examples. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph K 1 , 1 , 3 \ e obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph K 1 , 1 , 3 . The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices. [ABSTRACT FROM AUTHOR]

Let G be a graph with set of vertices V (G) (| V (G) | = n) and edge set E (G) . Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by S O (G) and is defined as S O = S O (G) = ∑ v i v j ∈ E (G) d G (v i) 2 + d G (v j) 2 , where d G (v i) is the degree of the vertex v i in G. In this paper we present some lower and upper bounds on the Sombor index of graph G in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

Let w G (u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, C W (G) , is the number of different values of w G (u) in G, and the eccentric complexity, C ec (G) , is the number of different eccentricities in G. In this paper, we prove that for every integer c there are infinitely many graphs G such that C W (G) − C ec (G) = c . Moreover, we prove this statement using graphs with the smallest possible cyclomatic number. That is, if c ≥ 0 we prove this statement using trees, and if c < 0 we prove it using unicyclic graphs. Further, we prove that C ec (G) ≤ 2 C W (G) − 1 if G is a unicyclic graph. In our proofs we use that the function w G (u) is convex on paths consisting of bridges. This property also promptly implies the already known bound for trees C ec (G) ≤ C W (G) . Finally, we answer in positive an open question by finding infinitely many graphs G with diameter 3 such that C ec (G) < C W (G) . [ABSTRACT FROM AUTHOR]

The crossing number cr (G) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n . [ABSTRACT FROM AUTHOR]

Recently, Mlaiki et al. introduced the notion of a controlled metric type space which is a generalization of the b-metric space. In this work, we define the controlled partial metric type space and give some fixed-point theorems for extensions of Kannan contraction in this space with suitable conditions. Moreover, as an application, we derive a fixed-point theorem for graphic contraction on the considered metric space endowed with a graph. [ABSTRACT FROM AUTHOR]

Determining vertex subsets are known tools to provide information about automorphism groups of graphs and, consequently about symmetries of graphs. In this paper, we provide both lower and upper bounds of the minimum size of such vertex subsets, called the determining number of the graph. These bounds, which are performed for arbitrary graphs, allow us to compute the determining number in two different graph families such are cographs and unit interval graphs. [ABSTRACT FROM AUTHOR]

The investigation of symmetric/asymmetric structures and their applications in mathematics (in particular in operator theory and functional analysis) is useful and fruitful. A metric space has the property of symmetry. By looking in the same direction and using the α -admissibility with regard to η and θ -functions, we demonstrate some existence and uniqueness fixed point theorems. The obtained results extend and generalize the main result of Isik et al. (2019). At the end, some illustrated applications are presented. [ABSTRACT FROM AUTHOR]

Fullerenes are molecules that can be presented in the form of cage-like polyhedra, consisting only of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex v of a graph is a local graph invariant defined as the sum of distances from v to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order n ≤ 232 and IPR fullerene graphs of order n ≤ 270 are presented. The structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed, and formulas for the Wiener index of several families of graphs are obtained. [ABSTRACT FROM AUTHOR]

We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs. [ABSTRACT FROM AUTHOR]

Let G 0 be a connected graph on n vertices and m edges. The R-graph R (G 0) of G 0 is a graph obtained from G 0 by adding a new vertex corresponding to each edge of G 0 and by joining each new vertex to the end points of the edge corresponding to it. Let G 1 and G 2 be graphs on n 1 and n 2 vertices, respectively. The R-graph double corona G 0 (R) ∘ { G 1 , G 2 } of G 0 , G 1 and G 2 , is the graph obtained by taking one copy of R (G 0) , n copies of G 1 and m copies of G 2 and then by joining the i-th old-vertex of R (G 0) to every vertex of the i-th copy of G 1 and the j-th new vertex of R (G 0) to every vertex of the j-th copy of G 2 . In this paper, we consider resistance distance in G 0 (R) ∘ { G 1 , G 2 } . Moreover, we give an example to illustrate the correction and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs G 1 , G 2 , ... , G k . Recall that H is an arbitrary graph with V (H) = { 1 , 2 , ... , k } , and G 1 , G 2 , ... , G k are disjoint graphs. Then, the H-join of graphs G 1 , G 2 , ... , G k , denoted by ⋁ H { G 1 , G 2 , ... , G k } , is a graph formed by taking G 1 , G 2 , ... , G k and joining every vertex of G i to every vertex of G j whenever i is adjacent to j in H. Here, we first give the Laplacian matrix of ⋁ H { G 1 , G 2 , ... , G k } , and then give a { 1 } -inverse L (⋁ H { G 1 , G 2 , ... , G k }) { 1 } or group inverse L (⋁ H { G 1 , G 2 , ... , G k }) # of L (⋁ H { G 1 , G 2 , ... , G k }) . It is well know that, there exists a relationship between resistance distance and entries of { 1 } -inverse or group inverse. Therefore, we can easily obtain resistance distance in ⋁ H { G 1 , G 2 , ... , G k } . In addition, some applications are presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of an α -admissible nonself-mapping, an α -F-contractive nonself-mapping, a weak α -F-contractive nonself-mapping, and a generalized α -F-contractive nonself-mapping and prove some P P F (past-present-future)-dependent fixed point theorems for the proposed contractive nonself-mappings in certain Razumikhin classes. By using our results, we derive some P P F -dependent fixed point theorems for an α -F-contractive nonself-mapping endowed with a graph or a partial order. Finally, we give some applications to illustrate the main results. [ABSTRACT FROM AUTHOR]