Applications of graph’s complete degree with bipolar fuzzy information

Due to the presence of two opposite directional thinking in relationships between countries and communication systems, the systems may not always be balanced. Therefore, the perfectness between countries relations are highly important. It comes from how much they were connected to each other for communication. In this study, first perfectly regular bipolar fuzzy graph is introduced and examined the regularity of nodes. Then, the relationship between the adjacent nodes and their regularity are visualized as a perfectly edge-regular bipolar fuzzy graphs. The totally accurate communication between all connected nodes is explained by introducing completely open neighborhood degree and completely closed neighborhood degree of nodes and edges in a bipolar fuzzy graph. Some algorithms and flowcharts of the proposed methods are given. Finally, two applications of these cogitation are exhibited in two bipolar fuzzy fields. The first one is in international relationships between some countries during cold-war era and the second one is in decision-making between teachers–students communication system for the improvement of teaching.

WEYL TYPE THEOREMS FOR ALGEBRIACALLY CLASS $p$-$wA(s,t)$ OPERATORS

In this paper, we study Weyl type theorems for $f(T)$, where $T$ is algebraically class $p$-$wA(s, t)$ operator with $0 < p \leq 1$ and $0 < s, t, s + t \leq 1$ and $f$ is an analytic function defined on an open neighborhood of the spectrum of $T$. Also we show that if $A , B^{*} \in B(\mathcal{H}) $ are class $p$-$wA(s, t)$ operators with $0 < p \leq 1$ and $0 < s, t, s + t \leq 1$,then generalized Weyl's theorem , a-Weyl's theorem, property $(w)$, property $(gw)$ and generalized a-Weyl's theorem holds for $f(d_{AB})$ for every $f \in H(\sigma(d_{AB})$, where $ d_{AB}$ denote the generalized derivation $\delta_{AB}:B(\mathcal{H})\rightarrow B(\mathcal{H})$ defined by $\delta_{AB}(X)=AX-XB$ or the elementary operator $\Delta_{AB}:B(\mathcal{H})\rightarrow B(\mathcal{H})$ defined by $\Delta_{AB}(X)=AXB-X$.

Unobstructedness of hyperkähler twistor spaces

A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

Recovery of radial-axial velocity in axisymmetric swirling flows of a viscous incompressible fluid in the Lagrangian consideration of vorticity evolution

Swirling laminar axisymmetric flows of viscous incompressible fluids in a potential field of body forces are considered. The study of flows is carried out in a cylindrical coordinate system. In the flows, the regions in which the axial derivative of the circumferential velocity cannot take on zero value in some open neighborhood (essentially swirling flows) and the regions in which this derivative is equal to zero (the region with layered swirl) are considered separately. It is shown that a well-known method (the method of viscous vortex domains) developed for non-swirling flows can be used for regions with layered swirling. For substantially swirling flows, a formula is obtained for calculating the radial-axial velocity of an imaginary fluid through the circumferential vorticity component, the circumferential circulation of a real fluid, and the partial derivatives of these functions. The particles of this imaginary fluid “transfer” vortex tubes of the radial-axial vorticity component while maintaining the intensity of these tubes, and also “transfer” the circumferential circulation and the product of the circular vorticity component by some function of the distance to the axis of symmetry. A non-integral method for reconstructing the velocity field from the vorticity field is proposed. It is reduced to solving a system of linear algebraic equations in two variables. The obtained result is proposed to be used to extend the method of viscous vortex domains to swirling axisymmetric flows.

The Star Degree Centrality Problem: A Decomposition Approach

We consider the problem of identifying the induced star with the largest cardinality open neighborhood in a graph. This problem, also known as the star degree centrality (SDC) problem, is shown to be [Formula: see text]-complete. In this work, we first propose a new integer programming (IP) formulation, which has a smaller number of constraints and nonzero coefficients in them than the existing formulation in the literature. We present classes of networks in which the problem is solvable in polynomial time and offer a new proof of [Formula: see text]-completeness that shows the problem remains [Formula: see text]-complete for both bipartite and split graphs. In addition, we propose a decomposition framework that is suitable for both the existing and our formulations. We implement several acceleration techniques in this framework, motivated by techniques used in Benders decomposition. We test our approaches on networks generated based on the Barabási–Albert, Erdös–Rényi, and Watts–Strogatz models. Our decomposition approach outperforms solving the IP formulations in most of the instances in terms of both solution time and quality; this is especially true for larger and denser graphs. We then test the decomposition algorithm on large-scale protein–protein interaction networks, for which SDC is shown to be an important centrality metric. Summary of Contribution: In this study, we first introduce a new integer programming (NIP) formulation for the star degree centrality (SDC) problem in which the goal is to identify the induced star with the largest open neighborhood. We then show that, although the SDC can be efficiently solved in tree graphs, it remains [Formula: see text]-complete in both split and bipartite graphs via a reduction performed from the set cover problem. In addition, we implement a decomposition algorithm motivated by Benders decomposition together with several acceleration techniques to both the NIP formulation and the existing formulation in the literature. Our experimental results indicate that the decomposition implementation on the NIP is the best solution method in terms of both solution time and quality.

Quadruple Roman Domination in Trees

This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)<k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

LOCALLY-BALANCED $k$-PARTITIONS OF GRAPHS

In this paper we generalize locally-balanced $2$-partitions of graphs and introduce a new notion, the locally-balanced $k$-partitions of graphs, defined as follows: a $k$-partition of a graph $G$ is a surjection $f:V(G)\rightarrow \{0,1,\ldots,k-1\}$. A $k$-partition ($k\geq 2$) $f$ of a graph $G$ is a locally-balanced with an open neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=i\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=j\}\vert \right\vert\leq 1.$$ A $k$-partition ($k\geq 2$) $f^{\prime}$ of a graph $G$ is a locally-balanced with a closed neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=i\}\vert - \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=j\}\vert \right\vert\leq 1.$$ The minimum number $k$ ($k\geq 2$), for which a graph $G$ has a locally-balanced $k$-partition with an open (a closed) neighborhood, is called an $lb$-open ($lb$-closed) chromatic number of $G$ and denoted by $\chi_{(lb)}(G)$ ($\chi_{[lb]}(G)$). In this paper we determine or bound the $lb$-open and $lb$-closed chromatic numbers of several families of graphs. We also consider the connections of $lb$-open and $lb$-closed chromatic numbers of graphs with other chromatic numbers such as injective and $2$-distance chromatic numbers.

ON LOCALLY-BALANCED 2-PARTITIONS OF BIPARTITE GRAPHS

A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood}, if for every $v\in V(G)$, $\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1$. A bipartite graph is \emph{$(a,b)$-biregular} if all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. In this paper we prove that the problem of deciding, if a given graph has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete even for $(3,8)$-biregular bipartite graphs. We also prove that a $(2,2k+1)$-biregular bipartite graph has a locally-balanced $2$-partition with an open neighbourhood if and only if it has no cycle of length $2 \pmod{4}$. Next, we prove that if $G$ is a subcubic bipartite graph that has no cycle of length $2 \pmod{4}$, then $G$ has a locally-balanced $2$-partition with an open neighbourhood. Finally, we show that all doubly convex bipartite graphs have a locally-balanced $2$-partition with an open neighbourhood.

Some Grüss Type Inequalities for Fréchet Differentiable Mappings

Let X be a Hilbert C∗-module on C∗-algebra A and p ∈ A. We denote by Dp(A,X) the set of all continuous functions f : A → X, which are Fréchet differentiable on a open neighborhood U of p. Then, we introduce some generalized semi-inner products on Dp(A,X), and using them some Grüss type inequalities in semi-inner product C∗-module Dp(A,X) and Dp(A,Xn) are established.

Secure w-Domination in Graphs

This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where w=(w0,w1,…,wl) is a vector of nonnegative integers such that w0≥1. The secure w-domination number is defined as follows. Let G be a graph and N(v) the open neighborhood of v∈V(G). We say that a function f:V(G)⟶{0,1,…,l} is a w-dominating function if f(N(v))=∑u∈N(v)f(u)≥wi for every vertex v with f(v)=i. The weight of f is defined to be ω(f)=∑v∈V(G)f(v). Given a w-dominating function f and any pair of adjacent vertices v,u∈V(G) with f(v)=0 and f(u)>0, the function fu→v is defined by fu→v(v)=1, fu→v(u)=f(u)−1 and fu→v(x)=f(x) for every x∈V(G)\{u,v}. We say that a w-dominating function f is a secure w-dominating function if for every v with f(v)=0, there exists u∈N(v) such that f(u)>0 and fu→v is a w-dominating function as well. The secure w-domination number of G, denoted by γws(G), is the minimum weight among all secure w-dominating functions. This paper provides fundamental results on γws(G) and raises the challenge of conducting a detailed study of the topic.