In this paper, we introduce the notion of normalized distance Laplacian matrices for signed graphs corresponding to the two signed distances defined for signed graphs. We characterize balance in signed graphs using these matrices and compare the normalized distance Laplacian spectral radius of signed graphs with that of all-negative signed graphs. Also we characterize the signed graphs having maximum normalized distance Laplacian spectral radius.

In this article, we study the signed distance matrix of the product of signed graphs such as the Cartesian product and the lexicographic product in terms of the signed distance matrices of the factor graphs. Also, we discuss the distance spectra of some special classes of product of signed graphs.

This paper deals with the problem of testing statistical hypotheses when the available data are fuzzy. In this approach, we first obtain a fuzzy test statistic based on fuzzy data, and then, based on a new signed distance between fuzzy numbers, we introduce a new decision rule to accept/reject the hypothesis of interest. The proposed approach is investigated for two cases: the case without nuisance parameters and the case with nuisance parameters. This method is employed to test the hypotheses for the mean of a normal distribution with known/unknown variance, the variance of a normal distribution, the difference of means of two normal distributions with known/unknown variances, and the ratio of variances of two normal distributions.

This paper aims to study the multi-objective transportation problem with fuzzy parameters. These fuzzy parameters represented as (α , β ) interval-valued fuzzy numbers instead of the normal fuzzy numbers. Using the signed distance ranking, the problem converted into the corresponding crisp multi-objective transportation problem. Then, the solution method introduced by [8] for solving the problem is applied. This method provides the ideal and the set of all (α , β ) fuzzy efficient solutions. The advantage of this method is more flexible than the standard multi-objective transportation problem, where it allows the decision maker to choose the (α , β ) levels of fuzzy numbers he is willing. A numerical example to illustrate the utility, effectiveness, and applicability of the method is given.

A signed graph is an ordered pair  = (G;  ); where G = (V; E) is the underlying graph of  with a signature function  : E! f1; 1g. In this article, we de ne the n th power of a signed graph and discuss some properties of these powers of signed graphs. As we can de ne two types of signed graphs as the power of a signed graph, necessary and su cient conditions are given for an n power of a signed graph to be unique. Also, we characterize balanced power signed graphs.

distance-based clustering methods categorize samples by optimizing a global criterion, finding ellipsoid clusters with roughly equal sizes. In contrast, density-based clustering techniques form clusters with arbitrary shapes and sizes by optimizing a local criterion. Most of these methods have several hyper-parameters, and their performance is highly dependent on the hyper-parameter setup. Recently, a Gaussian Density distance (GDD) approach was proposed to optimize local criteria in terms of distance and density properties of samples. GDD can find clusters with different shapes and sizes without any free parameters. However, it may fail to discover the appropriate clusters due to the interfering of clustered samples in estimating the density and distance properties of remaining unclustered samples. Here, we introduce Adaptive GDD (AGDD), which eliminates the inappropriate effect of clustered samples by adaptively updating the parameters during clustering. It is stable and can identify clusters with various shapes, sizes, and densities without adding extra parameters. The distance metrics calculating the dissimilarity between samples can affect the clustering performance. The effect of different distance measurements is also analyzed on the method. The experimental results conducted on several well-known datasets show the effectiveness of the proposed AGDD method compared to the other well-known clustering methods.

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.

A signed graph (or, in short, sigraph) S = (Su, s) consists of an underlying graph Su:= G = (V, E) and a function s: E (Su) ®{+, -}, called the signature of S. A marking of S is a function m: V (S) ®{+, -}. The canonical marking of a signed graph S, denoted ms, is given asms (v):= Õ vw2E(S) s (vw).The line graph of a graph G, denoted L (G), is the graph in which edges of G are represented as vertices, two of these vertices are adjacent if the corresponding edges are adjacent in G. There are three notions of a line signed graph of a signed graph S = (Su, s) in the literature, viz., L (S), Lx (S) and L· (S), all of which have L (Su) as their underlying graph, only the rule to assign signs to the edges of L (Su) differ. Every edge ee ′in L (S) is negative whenever both the adjacent edges e and e’ in S are negative, an edge ee′in Lx (S) has the product s (e) s (e′) as its sign and an edge ee′in L· (S) has ms (v) as its sign, where vÎ V (S) is a common vertex of edges e and e′.The line-cut graph (or, in short, lict graph) of a graph G = (V,E), denoted by Lc (G), is the graph with vertex set E (G) È C (G), where C (G) is the set of cut-vertices of G, in which two vertices are adjacent if and only if they correspond to adjacent edges of G or one vertex corresponds to an edge e of G and the other vertex corresponds to a cut-vertex c of G such that e is incident with c.In this paper, we introduce dot-lict signed graph (or ·-lict signed graph) L·c (S), which has Lc (Su) as its underlying graph. Every edge uv in L·c (S) has the sign ms (p), if u, v Î E (S) and p Î V (S) is a common vertex of these edges, and it has the sign ms (v), if u Î E (S) and v Î C (S). We characterize signed graphs on Kp, p³2, on cycle Cn and on Km, n which are ·-lict signed graphs or ·-line signed graphs, characterize signed graphs S so that L·c (S) and L· (S) are balanced. We also establish the characterization of signed graphs S for which S ~ L·c (S), S ~ L· (S), h (S) ~ L·c (S) and h (S) ~ L· (S), here h (S) is negation of S and ~ stands for switching equivalence.

The energy of signed graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two signed graphs are said to be equienergetic if they have same energy. In the literature the construction of equienergetic signed graphs are reported. In this paper we obtain the characteristic polynomial and energy of the join of two signed graphs and thereby we give another construction of unbalanced, noncospectral equieneregtic signed graphs on n  8 vertices.