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Motivated by the inverse formula of the distance matrix of a tree and the Moore–Penrose inverse of a circum-Euclidean distance matrix (CEDM), in this paper, we study a general real square matrix M whose Moore–Penrose inverse can be expressed as the sum of a Laplacian-like matrix L and a rank one matrix. In particular, for a symmetric hollow matrix M, under an assumption, we show that M is a Euclidean distance matrix if and only if L is positive semidefinite. Based on this, we obtain a new characterization for CEDMs involving their Moore–Penrose inverses. As an application, we show that the distance matrices of block graphs and odd-cycle-clique graphs are CEDMs. Finally, we establish an interlacing property between the eigenvalues of a Euclidean distance matrix M (including the singular case) and its associated Laplacian-like matrix L, which generalizes the interlacing property proved for the distance matrices of trees. [ABSTRACT FROM AUTHOR]

The paper analyzes interregional trade in Russia using gravity models. The model estimates the trade elasticity with respect to the size of exporting and importing regions and the distance between them. In addition, the impact on trade of additional factors, such as the common border of trading regions, the presence or absence of railroads, land or sea borders with other countries, is studied. Special attention is given to the issue of measuring distances between regions. The influence of the method of calculating the distance matrix (from the simplest orthodromic to the proposed weighted matrix of the shortest road and rail distances) on the coefficients of the models is studied. The all-Russian estimates of trade elasticities by the size of the exporting and importing region, equal to 1.15 and 1.05, showed high accuracy and robustness to the set of factors included in the model, the observation period, and the distance matrix. Both values were greater than one, which is significantly higher than typical estimates for international trade. This suggests that large and wealthy regions in Russia trade more, further increasing their welfare, while small and depressed regions are unable to escape the poverty trap, further increasing the current high level of regional heterogeneity. Distance is also very important in Russia (the elasticity of trade with respect to distance is –1.15, which is much higher than the world average, but still lower than the previous estimates for Siberia and the Russian Far East). This indicates insufficient transport infrastructure, higher costs of information search, transactions, contract execution, and other difficulties associated with long-distance trade. The absence of railroads in a region reduces its trade by about one-third, while neighboring regions increase the quantity of goods transported between them by about 75%. An external land or sea border facilitates domestic imports, some of which are re-exported abroad and some are consumed with the money earned from exports. At the same time, domestic exports from border regions, which cannot compete with external exports, are reduced. The method of calculating the distance matrix has a significant effect on the elasticity of trade with respect to distance, and to a limited extent on other coefficients of the model. In this case, it is recommended to use the weighted matrix proposed in this paper, which uses road distances for nearby regions and rail distances for distant regions

Formulas for the determinant of distance matrix DT{D}_{T} of tree TT are known in the unweighted case and in the case when the edges of TT have commuting variable weights. Associated with the four-point condition (4PC) and a tree TT are two matrices, the Max4PCT{{\rm{Max4PC}}}_{T} and the Min4PCT{{\rm{Min4PC}}}_{T}. These are not full rank matrices and their rank, a basis BB, and formulas for the determinant when restricted to the rows and columns of BB are known. In this work, we generalize both these matrices to the case when the edges of TT have commuting variable weights and determine edge-weighted counterparts of known results.

This research explores the energy of the prime ideal graph of a commutative ring. The study demonstrates the energy formula of the graph associated with transmission-based matrices. Through research, the findings highlight the distance, Wiener-Hosoya, and distance signless Laplacian matrices. It should be noted that the distance and Wiener-Hosoya energies are always twice their spectral radius, meanwhile, it does not hold for distance signless Laplacian energy. [ABSTRACT FROM AUTHOR]

Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study of graphs with "few eigenvalues" is a contemporary problem in spectral graph theory. This paper studies graphs with few distinct distance eigenvalues. After mentioning the classification of graphs with one and two distinct distance eigenvalues, we mainly focus on graphs with three distinct distance eigenvalues. Characterizing graphs with three distinct distance eigenvalues is "highly" non-trivial. In this paper, we classify all trees whose distance matrix has precisely three distinct eigenvalues. Our proof is different from earlier existing proof of the result as our proof is extendable to other similar families such as unicyclic and bicyclic graphs. The main tools which we employ include interlacing and equitable partitions. We also list all the connected graphs on ν ≤ 6 vertices and compute their distance spectra. Importantly, all these graphs on ν ≤ 6 vertices are determined from their distance spectra. We deliver a distance cospectral pair of order 7, thus making it a distance cospectral pair of the smallest order. This paper is concluded with some future directions. [ABSTRACT FROM AUTHOR]

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Sentiment analysis is a method of analyzing data to identify its intent. It identifies the emotional tone of a text body. Aspect-based sentiment analysis is a text analysis technique that identifies the aspect and the sentiment associated with each aspect. Different organizations use aspect-based sentiment analysis to analyze opinions about a product, service, or idea. Traditional sentiment analysis methods analyze the complete text and assign a single sentiment label to it. They do not handle the tasks of aspect association, dealing with multiple aspects and inclusion of linguistic concepts together as a system. In this article, AlgoDM, an algorithm to perform aspect-based sentiment analysis is proposed. AlgoDM uses a novel concept of IDistance matrix to extract aspects, associate aspects with sentiment words, and determine the sentiment associated with each aspect. The IDistance matrix is constructed to calculate the distance between aspects and the words expressing the sentiment related to the aspect. It works at the sentence level and identifies the opinion expressed on each aspect appearing in the sentence. It also evaluates the overall sentiment expressed in the sentence. The proposed algorithm can perform sentiment analysis of any opinionated text. [ABSTRACT FROM AUTHOR]

Almost Moore mixed graphs appear in the context of the degree/diameter problem as a class of extremal mixed graphs, in the sense that their order is one unit less than the Moore bound for such graphs. The problem of their existence has been considered just for diameter 2. In this paper, we give a complete characterization of these extremal mixed graphs for diameters 2 and 3. We also derive some optimal constructions for other diameters. [ABSTRACT FROM AUTHOR]

Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at ⌊ n 2 ⌋. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a simple graph and (R, +, ·) be a ring with zero element 0R. The Wiener index of G, denoted by W(G), is defined as the sum of distances of every vertex u and v, or half of the sum of all entries of its distance matrix. The prime graph of R, denoted by P G(R), is defined as a graph with V (P G(R)) = R such that uv ∈ E(P G(R)) if and only if uRv = {0R} or vRu = {0R}. In this article, we determine the Wiener index of P G(Zn) in some cases n by constructing its distance matrix. We partition the set Zn into three types of sets, namely zero sets, nontrivial zero divisor sets, and unit sets. There are two objectives to be achieved. Firstly, we revise the Wiener index formula of P G(Zn) for n = p² and n = p³ for prime number p and we compare this results with the results carried out by previous researchers. Secondly, we determine the Wiener index formula of P G(Zn) for n = pq, n = p² q, n = p² q², and n = pqr for distinct prime numbers p, q, and r. [ABSTRACT FROM AUTHOR]

For a commutative ring R with unity, the cozero-divisor graph denoted by Γ'(R), is an undirected simple graph whose vertex set is the set of all non-zero and non-unit elements of R. Two distinct vertices x and y are adjacent if and only if x does not belong to the ideal Ry and y does not belong to Rx. The cozero-divisor graph on the ring of integers modulo n is a generalized join of its induced sub graphs all of which are null graphs. This property of the cozero-divisor graph on Zn is used in finding its distance spectrum. In this paper, the distance matrix of the cozero-divisor graph on the ring of integers modulo n is discovered and the general method is discussed to find its distance spectrum, for any value of n. Also, the distance spectrum of this graph is explored for some values of n, by means of the vertex weighted distance matrix of the co-proper divisor graph of n. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with adjacency matrix $ A(G) $ A (G) and distance matrix $ \mathcal {D}(G) $ D (G). The adjacency-distance matrix of G is defined as $ S(G) = \mathcal {D}(G) + A(G) $ S (G) = D (G) + A (G). In this paper, $ S(G) $ S (G) is generalized by the convex linear combinations \[ S_{\alpha}(G)=\alpha \mathcal{D}(G)+(1-\alpha)A(G) \] S α (G) = α D (G) + (1 − α) A (G) where $ \alpha \in [0,1] $ α ∈ [ 0 , 1 ]. Let $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) be the spectral radius of $ S_{\alpha }(G) $ S α (G). This paper presents results on $ S_{\alpha }(G) $ S α (G) with emphasis on $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) and some results on $ S(G) $ S (G) are extended to all α in some subintervals of $ [0,1] $ [ 0 , 1 ]. For $ \alpha \in [1/2,1] $ α ∈ [ 1 / 2 , 1 ] , the trees attaining the largest and the smallest $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) among trees of fixed order are determined and it is proved that $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) is a branching index. Moreover, for $ \alpha \in (1/2,1] $ α ∈ (1 / 2 , 1 ] , the graphs that uniquely minimize $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) : among all connected graphs of fixed order and fixed connectivity, and among all connected graphs of fixed order and fixed chromatic number are characterized. [ABSTRACT FROM AUTHOR]

Distance or dissimilarity matrices are widely used in applications. We study the relationships between the eigenvalues of the distance matrices and outliers and show that outliers affect the pairwise distances and inflate the eigenvalues. We obtain the eigenvalues of a distance matrix that is affected by k outliers and compare them to the eigenvalues of a distance matrix with a constant structure. We show a discrepancy in the sizes of the eigenvalues of a distance matrix that is contaminated with outliers, present an algorithm and offer a new outlier detection method based on the eigenvalues of the distance matrix. We compare the new distance-based outlier technique with several existing methods under five distributions. The methods are applied to a study of public utility companies and gene expression data. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph of order n having Wiener index W(G). The distance, distance Laplacian and the distance signless Laplacian energies of G are respectively defined asDE(G)=∑i=1n|υiD|,DLE(G)=∑i=1n|υiL−Tr‾|andDSLE(G)=∑i=1n|υiQ−Tr‾|, where υiD,υiL and υiQ,1≤i≤n are respectively the distance, distance Laplacian and the distance signless Laplacian eigenvalues of G and Tr‾=2W(G)n is the average transmission degree. In this paper, we will study the relation between DE(G), DLE(G) and DSLE(G). We obtain some necessary conditions for the inequalities DLE(G)≥DSLE(G),DLE(G)≤DSLE(G),DLE(G)≥DE(G) and DSLE(G)≥DE(G) to hold. We will show for graphs with one positive distance eigenvalue the inequality DSLE(G)≥DE(G) always holds. Further, we will show for the complete bipartite graphs the inequality DLE(G)≥DSLE(G)≥DE(G) holds. We end this paper by computational results on graphs of order at most 6.

Abstract As the need for ecosystem biodiversity assessment increases within the climate crisis framework, more and more studies using spectral variation hypothesis (SVH) are proposed to assess biodiversity at various scales. The SVH implies optical diversity (also called spectral diversity) is driven by light absorption dynamics associated with plant traits (PTs) variability (which is an indicator of functional diversity) which is, in turn, determined by biodiversity. In this study, we examined the relationship between PTs variability, optical diversity and α‐ and β‐diversity at different taxonomic ranks at the Monte Bondone grasslands, Trentino province, Italy. The results of the study showed that the PTs variability, at the α scale, was not correlated with biodiversity. On the other hand, the results observed at the community scale (β‐diversity) showed that the variation of some of the investigated biochemical and biophysical PTs was associated with the β‐diversity. We used the Mantel test to analyse the relationship between the PTs variability and species β‐diversity. The results showed a correlation coefficient of up to 0.50 between PTs variability and species β‐diversity. For higher taxonomic ranks such as family and functional groups, a slightly higher Spearman's correlation coefficient of up to 0.64 and 0.61 was observed, respectively. The SVH approach was also tested to estimate β‐diversity and we found that spectral diversity calculated by Spectral Angle Mapper showed to be a better proxy of biodiversity in the same ecosystem where the spectral diversity approach failed to estimate α‐diversity. These findings suggest that optical and PTs diversity approaches can be used to predict species diversity in the grasslands ecosystem where the species turnover is high.

Abstract Protein-ligand binding affinity plays a pivotal role in drug development, particularly in identifying potential ligands for target disease-related proteins. Accurate affinity predictions can significantly reduce both the time and cost involved in drug development. However, highly precise affinity prediction remains a research challenge. A key to improve affinity prediction is to capture interactions between proteins and ligands effectively. Existing deep-learning-based computational approaches use 3D grids, 4D tensors, molecular graphs, or proximity-based adjacency matrices, which are either resource-intensive or do not directly represent potential interactions. In this paper, we propose atomic-level distance features and attention mechanisms to capture better specific protein-ligand interactions based on donor-acceptor relations, hydrophobicity, and $$\pi $$ π -stacking atoms. We argue that distances encompass both short-range direct and long-range indirect interaction effects while attention mechanisms capture levels of interaction effects. On the very well-known CASF-2016 dataset, our proposed method, named Distance plus Attention for Affinity Prediction (DAAP), significantly outperforms existing methods by achieving Correlation Coefficient (R) 0.909, Root Mean Squared Error (RMSE) 0.987, Mean Absolute Error (MAE) 0.745, Standard Deviation (SD) 0.988, and Concordance Index (CI) 0.876. The proposed method also shows substantial improvement, around 2% to 37%, on five other benchmark datasets. The program and data are publicly available on the website https://gitlab.com/mahnewton/daap. Scientific Contribution Statement This study innovatively introduces distance-based features to predict protein-ligand binding affinity, capitalizing on unique molecular interactions. Furthermore, the incorporation of protein sequence features of specific residues enhances the model’s proficiency in capturing intricate binding patterns. The predictive capabilities are further strengthened through the use of a deep learning architecture with attention mechanisms, and an ensemble approach, averaging the outputs of five models, is implemented to ensure robust and reliable predictions.

The Four point condition (4PC henceforth) is a well known condition characterising distances in trees T. Let w , x , y , z be four vertices in T and let d x , y denote the distance between vertices x , y in T. The 4PC condition says that among the three terms d w , x + d y , z , d w , y + d x , z and d w , z + d x , y the maximum value equals the second maximum value. We define an ( n 2 ) × ( n 2 ) sized matrix Max4PC T from a tree T where the rows and columns are indexed by size-2 subsets. The entry of Max4PC T corresponding to the row indexed by { w , x } and column { y , z } is the maximum value among the three terms d w , x + d y , z , d w , y + d x , z and d w , z + d x , y. In this work, we determine basic properties of this matrix like rank, give an algorithm that outputs a family of bases, and find the determinant of Max4PC T when restricted to our basis. We further determine the inertia and the Smith Normal Form (SNF) of Max4PC T. [ABSTRACT FROM AUTHOR]

For positive integers $ m_{1},m_{2},\ldots,m_{h} $ m 1 , m 2 , ... , m h , a generalized balanced tree $ T(m_{1},m_{2},\ldots,m_{h}) $ T (m 1 , m 2 , ... , m h) is a rooted tree of height h such that every vertex of depth i has $ m_{i+1} $ m i + 1 children, $ 0\leq i\leq ~h-1 $ 0 ≤ i ≤ h − 1. The distance matrix $ D(G) $ D (G) of a simple connected graph G of order n is an $ n \times n $ n × n matrix whose $ (i,j){{\rm th}} $ (i , j) th entry is the distance between $ i{{\rm th}} $ i th and $ j{{\rm th}} $ j th vertices. A connected graph G is called a k-partitioned transmission regular graph if there exists a vertex partition $ \{V_{1},V_{2},\ldots,V_{k}\} $ { V 1 , V 2 , ... , V k } of G so that for $ 1\leq i,j \leq k $ 1 ≤ i , j ≤ k , and $ x\in V_{i} $ x ∈ V i , $ \sum _{y\in V_{j}} d(x,y) $ ∑ y ∈ V j d (x , y) is a constant. Here we show that $ T(m_{1},m_{2},\ldots,m_{h}) $ T (m 1 , m 2 , ... , m h) is an $ (h+1) $ (h + 1) -partitioned transmission regular graph. We find an $ (h+1)\times (h+1) $ (h + 1) × (h + 1) matrix whose largest eigenvalue is the distance spectral radius of $ T(m_{1},m_{2},\ldots,m_{h}) $ T (m 1 , m 2 , ... , m h). We obtain the characteristic polynomial of $ D(T(m_{1},m_{2},\ldots,m_{h})) $ D (T (m 1 , m 2 , ... , m h)) in terms of that of the smaller matrices and give an idea to find the full spectrum. Moreover, we get that $ D(T(m_{1},m_{2},\ldots,m_{h})) $ D (T (m 1 , m 2 , ... , m h)) has $ -2 $ − 2 an eigenvalue with multiplicity at least $ m_{1}\cdots m_{h-1} (m_{h}-1) $ m 1 ⋯ m h − 1 (m h − 1) and $ -(m_{h}+2)\pm \sqrt {m_{h}(m_{h}+4)} $ − (m h + 2) ± m h (m h + 4) as eigenvalues with multiplicity at least $ m_{1}\cdots m_{h-2} (m_{h-1}-1) $ m 1 ⋯ m h − 2 (m h − 1 − 1). [ABSTRACT FROM AUTHOR]

As the need for ecosystem biodiversity assessment increases within the climate crisis framework, more and more studies using spectral variation hypothesis (SVH) are proposed to assess biodiversity at various scales. The SVH implies optical diversity (also called spectral diversity) is driven by light absorption dynamics associated with plant traits (PTs) variability (which is an indicator of functional diversity) which is, in turn, determined by biodiversity. In this study, we examined the relationship between PTs variability, optical diversity and α‐ and β‐diversity at different taxonomic ranks at the Monte Bondone grasslands, Trentino province, Italy. The results of the study showed that the PTs variability, at the α scale, was not correlated with biodiversity. On the other hand, the results observed at the community scale (β‐diversity) showed that the variation of some of the investigated biochemical and biophysical PTs was associated with the β‐diversity. We used the Mantel test to analyse the relationship between the PTs variability and species β‐diversity. The results showed a correlation coefficient of up to 0.50 between PTs variability and species β‐diversity. For higher taxonomic ranks such as family and functional groups, a slightly higher Spearman's correlation coefficient of up to 0.64 and 0.61 was observed, respectively. The SVH approach was also tested to estimate β‐diversity and we found that spectral diversity calculated by Spectral Angle Mapper showed to be a better proxy of biodiversity in the same ecosystem where the spectral diversity approach failed to estimate α‐diversity. These findings suggest that optical and PTs diversity approaches can be used to predict species diversity in the grasslands ecosystem where the species turnover is high. [ABSTRACT FROM AUTHOR]

Protein-ligand binding affinity plays a pivotal role in drug development, particularly in identifying potential ligands for target disease-related proteins. Accurate affinity predictions can significantly reduce both the time and cost involved in drug development. However, highly precise affinity prediction remains a research challenge. A key to improve affinity prediction is to capture interactions between proteins and ligands effectively. Existing deep-learning-based computational approaches use 3D grids, 4D tensors, molecular graphs, or proximity-based adjacency matrices, which are either resource-intensive or do not directly represent potential interactions. In this paper, we propose atomic-level distance features and attention mechanisms to capture better specific protein-ligand interactions based on donor-acceptor relations, hydrophobicity, and π -stacking atoms. We argue that distances encompass both short-range direct and long-range indirect interaction effects while attention mechanisms capture levels of interaction effects. On the very well-known CASF-2016 dataset, our proposed method, named Distance plus Attention for Affinity Prediction (DAAP), significantly outperforms existing methods by achieving Correlation Coefficient (R) 0.909, Root Mean Squared Error (RMSE) 0.987, Mean Absolute Error (MAE) 0.745, Standard Deviation (SD) 0.988, and Concordance Index (CI) 0.876. The proposed method also shows substantial improvement, around 2% to 37%, on five other benchmark datasets. The program and data are publicly available on the website https://gitlab.com/mahnewton/daap. Scientific Contribution Statement This study innovatively introduces distance-based features to predict protein-ligand binding affinity, capitalizing on unique molecular interactions. Furthermore, the incorporation of protein sequence features of specific residues enhances the model's proficiency in capturing intricate binding patterns. The predictive capabilities are further strengthened through the use of a deep learning architecture with attention mechanisms, and an ensemble approach, averaging the outputs of five models, is implemented to ensure robust and reliable predictions. [ABSTRACT FROM AUTHOR]

The paper introduces a novel probability descriptor for genome sequence comparison, employing a generalized form of Jensen-Shannon divergence. This divergence metric stems from a one-parameter family, comprising fractions up to a maximum value of half. Utilizing this metric as a distance measure, a distance matrix is computed for the new probability descriptor, shaping Phylogenetic trees via the neighbor-joining method. Initial exploration involves setting the parameter at half for various species. Assessing the impact of parameter variation, trees drawn at different parameter values (half, one-fourth, one-eighth). However, measurement scales decrease with parameter value increments, with higher similarity accuracy corresponding to lower scale values. Ultimately, the highest accuracy aligns with the maximum parameter value of half. Comparative analyses against previous methods, evaluating via Symmetric Distance (SD) values and rationalized perception, consistently favor the present approach's results. Notably, outcomes at the maximum parameter value exhibit the most accuracy, validating the method's efficacy against earlier approaches. [ABSTRACT FROM AUTHOR]

The distance energy of a connected graph G is the sum of absolute values of the eigenvalues of the distance matrix of G. In this paper, we study how the distance energy of the complete split graph G S (m , n) = K m + K ¯ n changes when an edge is deleted from it. The complete split graph G S (m , n) consists of a clique on m vertices and an independent set on n vertices in which each vertex of the clique is adjacent to each vertex of the independent set. We prove that the distance energy of the complete split graph G S (m , n) always increases when an edge is deleted from it. [ABSTRACT FROM AUTHOR]

A conjugate skew gain graph is a skew gain graph with the labels (also called, the conjugate skew gains) from the field of complex numbers on the oriented edges such that they get conjugated when we reverse the orientation. In this paper, we introduce distance matrices for conjugate skew gain graphs and characterize balanced conjugate skew gain graphs using these matrices. We provide explicit formulae for the distance spectra of certain conjugate skew gain graphs. [ABSTRACT FROM AUTHOR]

The quadratic embedding constant (QEC) of a finite, simple, connected graph G is the maximum of the quadratic form of the distance matrix of G on the subset of the unit sphere orthogonal to the all-ones vector. The study of these QECs was motivated by the classical work of Schoenberg on quadratic embedding of metric spaces [ Ann. of Math. , 1935] and [ Trans. Amer. Math. Soc. , 1938]. In this article, we provide sharp upper and lower bounds for the QEC of trees. Next, we identify a new subclass of nonsingular graphs whose QEC is the second largest distance eigenvalue. Finally, we show that the QEC of the cluster of an arbitrary graph G with either a complete or star graph can be computed in terms of the QEC of G. As an application of this result, we provide new families of examples of graphs of QE class. [ABSTRACT FROM AUTHOR]

The article develops methods to solve a fundamental problem in computer vision: image recognition of visual objects. The results of the research on the construction of modifications for the space of classification features based on the application of the transformation of the structural description through the decomposition in the orthogonal basis and the implementation of the distance matrix model between the components of the description are presented. The application of the system of orthogonal functions as an apparatus for the transformation of the description showed the possibility of a significant gain in the speed of processing while maintaining high indicators of classification accuracy and interference resistance. The synthesized feature systems' effectiveness has been confirmed in terms of a significant increase in the rate of codes and a sufficient level of efficiency. An experimental example showed that the time spent calculating the relevance of descriptions according to their modified presentation is more than ten times shorter than for traditional metric approaches. The developed classification features can be used in applied tasks where the time of visual objects' identification is critical. [ABSTRACT FROM AUTHOR]

Let X = {x1,...,xn} be a finite set endowed with a metric d . The matrix A = (d(xi, xj))nxn is called a distance matrix. In this paper we discuss about the minimum rank that can be achieved by an n x n distance matrix and prove that the rank of every 5 x 5 and 6 x 6 distance matrix is not less than 4. [ABSTRACT FROM AUTHOR]

Aiming at the defects of the existing probabilistic hesitation fuzzy distance measures, which require the number of membership degree to be consistent and the order to be rearranged, a probabilistic hesitation fuzzy multi-attribute decision making method based on the comprehensive characteristic distance measure is proposed. First, a new law of probability hesitant fuzzy number comparison is defined. Then four characteristics of aggregation, discreteness, fuzziness and consistency are defined, and a new comprehensive distance measure is defined based on definitions above. Finally, on the basis of traditional TODIM method and prospect theory, the validity and rationality of the distance measure and new probability hesitant fuzzy recognition method in this paper are verified by examples and comparative analysis.

In this paper, we determine the distance Laplacian and distance signless Laplacian spectrum of generalized wheel graphs and a new class of graphs called dumbbell graphs.

Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study of graphs with “few eigenvalues” is a contemporary problem in spectral graph theory. This paper studies graphs with few distinct distance eigenvalues. After mentioning the classification of graphs with one and two distinct distance eigenvalues, we mainly focus on graphs with three distinct distance eigenvalues. Characterizing graphs with three distinct distance eigenvalues is “highly” non-trivial. In this paper, we classify all trees whose distance matrix has precisely three distinct eigenvalues. Our proof is different from earlier existing proof of the result as our proof is extendable to other similar families such as unicyclic and bicyclic graphs. The main tools which we employ include interlacing and equitable partitions. We also list all the connected graphs on ν ≤ 6 vertices and compute their distance spectra. Importantly, all these graphs on ν ≤ 6 vertices are determined from their distance spectra. We deliver a distance cospectral pair of order 7, thus making it a distance cospectral pair of the smallest order. This paper is concluded with some future directions.

AbstractLet G be a connected simple graph with n vertices. The distance Laplacian matrix [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the diagonal matrix of vertex transmissions and [Formula: see text] is the distance matrix of G. The eigenvalues of [Formula: see text] are the distance Laplacian eigenvalues of G and are denoted by [Formula: see text]. The largest eigenvalue [Formula: see text] is called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018), and Ma et al. (2018) completely characterized the graphs having some distance Laplacian eigenvalue of multiplicity [Formula: see text]. In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity [Formula: see text] together with one of the distance Laplacian eigenvalues as n of multiplicity either 3 or 2. Further, we completely determine the graphs for which the distance Laplacian eigenvalue n is of multiplicity [Formula: see text].

For a bipartite graph, the complete adjacency matrix is not necessary to display its adjacency information. In 1985, Godsil used a smaller size matrix to represent this, known as the bipartite adjacency matrix. Recently, the bipartite distance matrix of a tree with perfect matching was introduced as a concept similar to the bipartite adjacency matrix. It has been observed that these matrices are nonsingular, and a combinatorial formula for their determinants has been derived. In this article, we provide a combinatorial description of the inverse of the bipartite distance matrix and establish identities similar to some well-known identities. The study leads us to an unexpected generalization of the usual Laplacian matrix of a tree. This generalized Laplacian matrix, which we call the bipartite Laplacian matrix, is usually not symmetric, but it shares many properties with the usual Laplacian matrix. In addition, we study some of the fundamental properties of the bipartite Laplacian matrix and compare them with those of the usual Laplacian matrix.

Let G be a connected graph with n vertices, m edges. The distance signless Laplacian matrix DQ(G) is defined as DQ(G) = Diag(Tr(G)) + D(G), where Diag(Tr(G)) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The distance signless Laplacian eigenvalues of G are the eigenvalues of DQ(G) and are denoted by δ1Q(G), δ2Q(G), ..., δnQ(G). δ1Q is called the distance signless Laplacian spectral radius of DQ(G). In this paper, we obtain upper and lower bounds for SDQ (G) in terms of the Wiener index, the transmission degree and the order of the graph.

In this article, we extend the concept of distance energy for hypergraphs. We first establish a relation between the distance energy and the distance spectral radius. Then, we obtain some bounds for the distance energy in terms of some invariant of hypergraphs such as the determinant of the distance matrix, number of vertices, and Wiener index along with the distance energy of join of kk-uniform hypergraphs. Furthermore, it is shown that the determinant of the distance matrix of kk-uniform hyperstar on nn vertices is (−1)n−1(n−1)kn−kk−1{\left(-1)}^{n-1}\left(n-1){k}^{\tfrac{n-k}{k-1}}. Later, the distance spectrum of kk-uniform hyperstar is obtained, which gives the explicit distance energy of kk-uniform hyperstar.

In this paper, we determine the distance Laplacian and distance signless Laplacian spectrum of generalized wheel graphs and a new class of graphs called dumbbell graphs. [ABSTRACT FROM AUTHOR]

Phylogenetics is now fundamental in life sciences, providing insights into the earliest branches of life and the origins and spread of epidemics. However, finding suitable phylogenies from the vast space of possible trees remains challenging. To address this problem, for the first time, we perform both tree exploration and inference in a continuous space where the computation of gradients is possible. This continuous relaxation allows for major leaps across tree space in both rooted and unrooted trees, and is less susceptible to convergence to local minima. Our approach outperforms the current best methods for inference on unrooted trees and, in simulation, accurately infers the tree and root in ultrametric cases. The approach is effective in cases of empirical data with negligible amounts of data, which we demonstrate on the phylogeny of jawed vertebrates. Indeed, only a few genes with an ultrametric signal were generally sufficient for resolving the major lineages of vertebrates. Optimization is possible via automatic differentiation and our method presents an effective way forward for exploring the most difficult, data-deficient phylogenetic questions. [ABSTRACT FROM AUTHOR]

Suppose G is a connected simple graph with the vertex set V (G) = { v 1 , v 2 , ⋯ , v n } . Let d G (v i , v j) be the distance between v i and v j . Then the distance matrix of G is D (G) = (d ij) n × n , where d ij = d G (v i , v j) . Since D(G) is a non-negative real symmetric matrix, its eigenvalues can be arranged λ 1 (G) ≥ λ 2 (G) ≥ ⋯ ≥ λ n (G) , where eigenvalues λ 1 (G) and λ n (G) are called the distance spectral radius and the least distance eigenvalue of G, respectively. In this paper, we characterize the unique graph whose distance spectral radius attains maximum and minimum among all complements of graphs with two pendent vertices, respectively. Furthermore, we determine the unique graph whose least distance eigenvalue attains minimum among all complements of graphs with two pendent vertices. [ABSTRACT FROM AUTHOR]

For a connected simple graph G, the generalized distance matrix is defined by D α : = α T r (G) + (1 - α) D (G) , 0 ≤ α ≤ 1 , where Tr(G) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix. For particular values of α , we obtain the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix and other uncountable distance based matrices. Let ∂ 1 ≥ ∂ 2 ≥ ⋯ ≥ ∂ n be the D α eigenvalues of G and p ≥ 2 a real number, the Schatten p-norm is the p-th root of the sum of p-th powers of eigenvalues of D α , α ∈ [ 1 2 , 1 ] , that is, ‖ D α ‖ p p = ∂ 1 p + ∂ 2 p + ⋯ + ∂ n p . In this paper, we obtain various bounds for ‖ D α ‖ p p in terms of different graph parameters and characterize the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

Suppose G is a connected simple graph with the vertex set V (G) = { v 1 , v 2 , ⋯ , v n } . Let d G (v i , v j) be the least distance between v i and v j in G. Then the distance matrix of G is D (G) = (d ij) n × n , where d ij = d G (v i , v j) . Since D(G) is a non-negative real symmetric matrix, its eigenvalues can be arranged λ 1 (G) ≥ λ 2 (G) ≥ ⋯ ≥ λ n (G) , where eigenvalues λ 1 (G) and λ n (G) are called the distance spectral radius and the least distance eigenvalue of G, respectively. In this paper, we characterize the unique graph whose distance spectral radius attains maximum and minimum among all complements of graphs of diameter greater than three, respectively. Furthermore, we determine the unique graph whose least distance eigenvalue attains minimum among all complements of graphs of diameter greater than three. [ABSTRACT FROM AUTHOR]

In this correspondence, we introduced the concept of minimum roman dominating distance energy ERDd(G) of a graph G and computed minimum roman dominating distance energy of some standard graphs. Also, we discussed the properties of eigenvalues of a minimum roman dominating distance matrix ARDd(G). Finally, we derived the upper and lower bounds for ERDd(G). [ABSTRACT FROM AUTHOR]

A cactus is a connected graph in which any two cycles have at most one common vertex. We determine the unique graph that maximizes the distance spectral radius over all cacti with fixed numbers of vertices and cycles, and thus prove a conjecture on the distance spectral radius of cacti in Bose et al. [4]. We prove the result in the context of hypertrees. [ABSTRACT FROM AUTHOR]

AbstractStress is a centrality measure determined by the shortest paths passing through the given vertex. Noting that adjacency matrix playing an important role in finding the distance and the number of shortest paths between given pair of vertices, an interesting expression and also an algorithm are presented to find stress using adjacency matrix. The results and algorithm are suitably adopted to obtain betweenness centrality measure. Further results are extended to the cases of Cartesian product [Formula: see text] of graphs, corona graph [Formula: see text] and their special cases.