Journal of Algorithms and Computation
Volume:50 Issue: 1, Jun 2018

In 1994, degree distance  of a graph was introduced by Dobrynin, Kochetova and Gutman. And Gutman proposed the Gutman index of a graph in 1994. In this paper, we introduce the concepts of  multiplicative version of degree distance and the multiplicative version of Gutman index of a graph. We find the sharp upper bound for the  multiplicative version of degree distance and multiplicative version of  Gutman index of cartesian product of two connected graphs. And we compute the exact value for the cartesian product of two complete graphs. Using this result, we prove that our bound is tight. Also, we obtain the sharp upper bound for the multiplicative version of degree distance and the multiplicative version of Gutman index of strong product of connected and complete graphs. And we observe the exact value for the strong product of two complete graphs. From this,  we prove that our bound is tight.

Community detection has a wide variety of  applications in different fields such as data mining, social network analysis and so on. Label Propagation Algorithm (LPA) is a simple and fast community detection algorithm, but it has low accuracy. There have been presented   some advanced versions of LPA  in recent years such as CenLP and WILPAS. In this paper, we present  improved versions of CenLP and WILPAS methods called CenLP+ and WILPAS+ respectively.   Experiments and benchmarks demonstrate that while CenLP+ is as fast as CenLP, it outperforms CenLP on both synthetic and  real-world networks.  Moreover, while accuracy of WILPAS+ on synthetic networks  comparable with that of WILPAS, on real-world networks, WILPAS+ excels WILPAS. In addition,  whereas both presented methods CenLP+ and WILPAS+ show high accuracy on synthetic networks, on real-world networks they outperform remarkably all other tested label propagation based algorithms for  community detection. Therefore, since CenLP+ and WILPAS+ are both fast and accurate,  specially on real-world networks, they can efficiently reveal   community structures of mega-scale  social networks.

PCIPunderline{ }Tree is a Parallelized memory Confidentiality and Integrity Protection technology (PCIP) method that pre-diffuses and encrypts the data stored in off-chip counter value for confidentiality and integrity protection of data, adding redundant checking data to the protection data block to ensure the integrity and confidentiality of the memory data. Then use the Simple Scalar tool to run 10 processors (SPEC2000) benchmark programs to simulation test. The result shows that the PCIPunderline{ }Tree method is greatly improved the running efficiency and the efficiency of encryption methods to compare with the Parallelized Encryption and Integrity Checking Engine (PE-ICE) method, at the same time, through the redundant data to check data integrity is better than using complex Hash algorithm to calculate the checking value and reduce system delay, reduce storage overhead chip, and ensure the real-time encryption and checking.

Combinatorial filters, which are discrete representations of estimationprocesses, have been the subject of increasing interest from the roboticscommunity in recent years.%This paper considers automatic reduction of combinatorial filters to a givensize, even if that reduction necessitates changes to the filter's behavior.%We introduce an algorithmic problem called emph{improper filter reduction}, in which the input is a combinatorial filter F alongwith an integer k representing the target size.  The output is anothercombinatorial filter F' with at most k states, such that the differencein behavior between F and F' is minimal.We present two methods for measuring the distance between pairs of filters, describe dynamic programming algorithms for computing these distances, andshow that improper filter reduction is NP-hard under these methods.%We then describe two heuristic algorithms for improper filter reduction, onechanged{greedy sequential} approach, and one randomized global approach based on prior workon weighted improper graph coloring.  We have implemented these algorithmsand analyze the results of three sets of experiments.

A graph G is said to be one modulo three geometric mean graph if there is an injective function phi from the vertex set of G to the set {a mid 1leq a leq 3q-2} and either aequiv 0(mod 3) or aequiv 1(mod 3)} where q is the number of edges of G and phi induces a bijection phi^{*} form the edge set of G to {a mid 1leq aleq 3q-2 and aequiv 1(mod3)} given by phi^{*}(uv)=leftlceil sqrt{phi(u)phi(v)}rightrceil or leftlfloor sqrt{phi(u)phi(v)}rightrfloor and the function phi is called one modulo three geometric mean labeling of G. In this paper, we establish that some families of graphs admit one modulo three geometric mean labeling.the formula is not displayed correctly!

The analysis of vulnerability in networks generally involves some questionsabout how the underlying graph is connected. One is naturally interestedin studying the types of disruption in the network that maybe causedby failures of certain links or nodes. In terms of a graph, the concept ofconnectedness is used in different forms to study many of the measuresof vulnerability. When certain vertices or edges of a connected graphare deleted, one wants to know whether the remaining graph is stillconnected, and if so, what its vertex - or edge - connectivity is. If on theother hand, the graph is disconnected, the determination of the number ofits components or their orders is useful. Our purpose here is to describeand analyses the current status of the vulnerability measures, identify itsmore interesting variants, and suggest a most suitable measure ofvulnerability.

In a pointed conflict-free partial   (PCFP) colouring of a digraph, each vertex has at least  one in-neighbour with unique colour.  In this paper,  it is proved that PCFP k-colourability of digraphs is NP-complete,  for any k >0.  Nevertheless  for   paths and cycles, one can  in linear time  find  a PCFP colouring  with a minimum number of colours and for a given tree, one can find a PCFP 2-colouring. In this paper a bipartite digraph  whose arcs start from the same part is called  a one-way bipartite digraph. It is proved every one-way bipartite  planar digraph has a PCFP 6-colouring,  every one-way bipartite planar digraph whose each vertex  has in-degree zero or greater than one, has a  PCFP 5-colouring and  every one-way bipartite planar digraph whose each vertex  has in-degree zero or greater than two, has a  PCFP 2-colouring. Two simple algorithms are proposed for finding a  PCFP colouring of a given digraph  such that the number of colours used is not  more than the maximum out-degree of  the vertices. For a digraph with a given PCFP colouring,  it is shown how to recolour the vertices after vertex or arc insertion or deletion to obtain a PCFP colouring for the new digraph.

Tipping Point refers to the moment when an adaption or infection sustains itself in network without further external inputs. Until now, studies have mainly focused on the occurrence of the Tipping Point and what it leads to rather than what precedes it. This paper explores the situation leading to the Tipping Point during a process of diffusion in networks. The core of the debate is to manifest that the process can be introduced as an example of conditionally convergent series and that determining the tipping points’ occurrence is conditional to the arrangement of the series based on Reimann Rearrangement Theorem. Accordingly, the occurrence of curve does not follow a general formulation. That is called indeterminacy since that the predictions about tipping points for any diffusion over the network may include a variety of right answers, although such indeterminacy neither means there is no tipping point nor many.

In this paper we introduce a new graph labeling method called k-Total prime cordial. Let G be a (p,q) graph. Let f:V(G)to{1,2, ldots, k} be a map where k in mathbb{N} and k>1. For each edge uv, assign the label gcd(f(u),f(v)). f is called k-Total prime cordial labeling of G if left|t_{f}(i)-t_{f}(j)right|leq 1, i,j in {1,2, ldots, k} where t_{f}(x) denotes the total number of vertices and the edges labeled with x. We investigate k-total prime cordial labeling of some graphs and study the 4-total prime cordial labeling of path, cycle, complete graph etc.the formula is not displayed correctly!

This paper presents a new mathematical technique for computation of geometric series and summability. This technique uses Annamalai’s computing model of algorithmic geometric series and its mathematical structures for further development of the infinite geometric series and summability. This could be very interesting and informative for current students and researchers.In the research study, a novel technique has been presented for formation and computation of infinite geometric series and summability.

Yager family of t-norms is a parametric family of continuous nilpotent t-norms which is also one of the most frequently applied one. This family of t-norms is strictly increasing in its parameter and covers the whole spectrum of t-norms when the parameter is changed from zero to infinity. In this paper, we study a nonlinear optimization problem where the feasible region is formed as a system of fuzzy relational equations (FRE) defined by the Yager t-norm. We firstly investigate the resolution of the feasible region when it is defined with max-Yager composition and present some necessary and sufficient conditions for determining the feasibility and some procedures for simplifying the problem. Since the feasible solutions set of FREs is non-convex and the finding of all minimal solutions is an NP-hard problem, conventional nonlinear programming methods may involve high computation complexity. For these reasons, a method is used, which preserves the feasibility of new generated solutions. The proposed method does not need to initially find the minimal solutions. Also, it does not need to check the feasibility after generating the new solutions. Moreover, we present a technique to generate feasible max-Yager FREs as test problems for evaluating the performance of the current algorithm. The proposed method has been compared with Lu and Fang’s algorithm. The obtained results confirm the high performance of the proposed method in solving such nonlinear problems.

A mapping  f: V(G)rightarrowleft{0, 1, 2 right} is called 3-product cordial labeling if vert v_f(i)-v_f(j)vert leq 1 and vert e_f(i)-e_f(j)vert leq 1 for any i, jin {0, 1, 2}, where v_f(i) denotes the number of vertices labeled  with i, e_f (i) denotes the number of edges xy with f(x)f(y)equiv i(mod 3). A graph with  3-product cordial labeling is called  3-product cordial graph. In this paper we establish that vertex switching of wheel,gear graph and degree splitting of bistar are 3-product cordial graphs.the formula is not displayed correctly!