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One of the alternative media, which is potential to be developed is the use of natural fiber materials from the environment (indigenous material). This natural fiber in foreign country such as Thailand, Japan, United States has crossed the threshold of the quality as a work of art that create by an artist. Meanwhile in Indonesia it is still brand new to be develop as a media of art expression. Indeed, art paper has already developed in Indonesia, but it is still a boundary production of pieces of paper that will be directly applied as a handicraft media. Moreover, the preparation process is still using the conventional method using chemical materials such as NaOH, Hypochloride, Boric acid, Chlorin, H2O2, etc., which burden the environment. Therefore an alternative, environmental friendly process is necessary to be developed to exploit natural fiber materials as a media of art expression. For this reason in this project a research on fiber-pulping process using xylanase enzyme has been carried out. The use of xylanase is enable to decrease chlorin consumption and reduce chloro-organic concentration in pulp and waste, that also means a reduction in production cost. The pulp was then used to make a paper to be tested for its quality by the Center for Pulp and Paper (Balai Besar Pulp dan Kertas, BBPK). This research resulted in a) enzyme formulations for the fiber-pulping and preparation from variety of fiber materials: banana’s stalk, the abacca, hyacinth, straw, pineapple fiber and gold cocoon, b) data of paper quality, and c) work of arts from fiber/pulp material. In conclusion, enzyme can be applied in fiber-pulping process for lower impact on the environment and the workers.

A bipartite graph G with partite sets X and Y is called consecutively super edge-magic if there exists a bijective function f : V(G) ⋃ E(G) → {1,2,...,|V(G)| + |E(G)|} with the property that f(X) = {1,2,...,|X|}, f(Y) = {|X|+1, |X|+2,...,|V(G)|} and f(u)+f(v) +f(uv) is constant for each uv ∈ E(G). The question studied in this paper is for which bipartite graphs it is possible to add a finite number of isolated vertices so that the resulting graph is consecutively super edge-magic. If it is possible for a bipartite graph G, then we say that the minimum such number of isolated vertices is the consecutively super edge-magic deficiency of G; otherwise, we define it to be +∞. This paper also includes a detailed discussion of other concepts that are closely related to the consecutively super edge-magic deficiency.

V.G. Vizing showed that any graph belongs to one of two classes: Class 1 if χʹ(G) = Δ(G) or in class 2 if χʹ(G) = Δ(G) + 1, where χʹ(G) and Δ(G) denote the edge chromatic index of G and the maximum degree of G, respectively. This paper addresses the problem of finding the edge chromatic surplus of a cubic graph G in Class 2, namely the minimum cardinality of colour classes over all 4-edge chromatic colourings of E(G). An approach to face this problem - using a new parameter q - is given in [1]. Computing q is difficult for the general case of graph G, so there is the need to find restricted classes of G, where q is easy to compute. Working in the same sense as in this paper we give an upper bound of the edge chromatic surplus for a special type of cubic graphs, that is the class of bridgeless non-planar cubic graphs in which in each pair of crossing edges, the crossing edges are adjacent to a third edge.

In the present paper, we study a certain class of meromorphic univalent functions f(z) dened by the linear operator L(α,β) f (z). The aim of the present paper is to prove some properties for the class Σα,β,kλ;(h) to satisfy the certain subordination. AMS Subject Classification: 30C45.

This paper introduces continuous-like linear operators on bilinear spaces as a generalization of continuous linear operators on Hilbert spaces. It is well known that the existence of the adjoint of a linear operator on a Hilbert space is equivalent to the operator being continuous. In this paper , this result is extended to the class of linear operators on bilinear spaces. It is shown that the existence of the adjoint of a linear operator on a bilinear space is guaranteed if and only if the operator is continuous-like.

This paper describes a dissipative controller desig n for networked control systems modeled by the Markovian jump system. Dissipativity refers to the existence of a supply rate dealing with the sys tem such that the closed-loop system has a dissipative property. The main result of this paper is a solvability condition in terms of linear matrix inequalities fo r achieving dissipativity of a dynamic output feedback problem in networked control systems via the Markovian jump system approach. A numerical example is given to show the efficacy of the proposed design.

This paper describes the performance of a simple counter based entropy coder, as compared to other entropy coders, especially Huffman coder. Lossless data compression, such as Huffman coder and arithmetic coder, are designed to perform well over a wide range of data entropy. As a result, the coders require significant computational resources that could be the bottleneck of a compression implementation performance. In contrast, counter-based coders are designed to be optimal on a limited entropy range only. This paper shows the encoding and decoding process of counter-based coder can be simple and fast, very suitable for hardware and software implementations. It also reports that the performance of the designed coder is comparable to that of a much more complex Huffman coder.

Block cipher analysis covers randomness analysis and cryptanalysis. This paper proposes a new method potentially used for randomness analysis and cryptanalysis. The method uses true random sequence concept as a reference for measuring randomness level of a random sequence. By using this concept, this paper defines bias which represents violation of a random sequence from true random sequence. In this paper, block cipher is treated as a mapping function of a discrete time dynamical system. The dynamical system framework is used to make the application of various analysis techniques developed in dynamical system field becomes possible. There are three main parts of the methodology presented in this paper: the dynamical system framework for block cipher analysis, a new chaos game scheme and an extended measure concept related to chaos game and fractal analysis. This paper also presents the general procedures of the proposed method, which includes: symbolic dynamic analysis of discr ete dynamical system whose block cipher as its mapping function, random sequence construction, the random sequence usage as input of a chaos game scheme, output measurement of chaos game scheme using extended measure concept, analysis the result of the measurement. The analysis process and of a specific real or sample block cipher and the analysis result are beyond the scope of this paper.

This paper describes a novel procedure for determining the standard value of acquisition distortion of fingerprint images. Knowledge about the standard value of acquisition distortion of the fingerprint images is very important in determining the method for improving image quality. In this paper, we propose a model to determine the standard value that can be used in classifying the type of distortion of the fingerprint images based on the image quality. The results show that the standard value of acquisition distortion of the fingerprint images based on the image quality have values of the local clarity scores (LCS) follows: dry parameter values are in the range of 0.0127-0.0149, neutral parameter values are less than 0.0127, and oily parameter values are greater than 0.0149. Meanwhile, the global clarity scores (GCS) are as follows: dry parameter values are in the range of 0.0117-0.0120, neutral parameter values are less than 0.0117, and oily parameter values are greater than 0.0120; and ridge-valley thickness ratios (RVTR) are as follows: dry parameter values are less than 7.75E-05, neutral parameter values are 7.75E-05-5.94E-05, and oily parameter values are greater than 5.94E-05.

Deep learning techniques have been used effectively for rubber crop yield prediction. A hybrid of Convolutional Neural Network (CNN) and Recurrent Neural Network (RNN) is the best technique for crop yield prediction because it can effectively handle uncertainty of features. Hence, in this paper, a hybrid CNN-RNN method is proposed to forecast Hevea yields based on environmental data in Kerala state, India. The proposed hybrid CNN-RNN method reduces the internal covariate shift of CNN by batch normalization and solves the gradient vanishing or exploding problem of RNN using LSTM with a cell activation mechanism. The proposed method has three essential characteristics: (i) it captures the time dependency of environmental factors and improves the inherent computational time; (ii) it is capable of generalizing the yield prediction under uncertain conditions without loss of prediction accuracy; (iii) combined with the back propagation and feed forward method it can reveal the extent to which samples of weather conditions and soil data conditions are suitable to provide a clear boundary between rubber yield variations.

Dengue is one of the major health problems in Bangladesh and many people are died in recent years due to the severity of this disease. Therefore, in this paper, a SIRS model for the human and SI model for vector population with saturated incidence rate and constant treatment function has been presented to describe the transmission of dengue. The equilibrium points and the basic reproduction number have been computed. The conditions which lead the disease free equilibrium and the endemic equilibrium have been determined. The local stability for the equilibrium points has been established based on the eigenvalues of the Jacobian matrix and the global stability has been analyzed by using the Lyapunov function theory. It is found that the stability of equilibrium points can be controlled by the reproduction number. In order to calculate the infection rate, data for infected human populations have been collected from several health institutions of Bangladesh. Numerical simulations of various compartments have been generated using MATLAB to investigate the influence of the key parameters for the transmission of the disease and to support the analytical results. The effect of treatment function over the infected compartment has been illustrated. The sensitivity of the reproduction number concerning the parameters of the model has been analyzed. Finally, the most sensitive parameter that has the highest effect over reproduction number has been identified.

We consider a SEIR model for the spread (transmission) of an infectious disease. The model has played an important role due to world pandemic disease spread cases. Our contributions in this paper are three folds. Our first contribution is to provide successive approximation and variational iteration methods to obtain analytical approximate solutions to the SEIR model. Our second contribution is to prove that for solving the SEIR model, the variational iteration and successive approximation methods are identical when we have some particular values of Lagrange multipliers in the variational iteration formulation. Third, we propose a new multistage-analytical method for solving the SEIR model. Computational experiments show that the successive approximation and variational iteration methods are accurate for small size of time domain. In contrast, our proposed multistage-analytical method is successful to solve the SEIR model very accurately for large size of time domain. Furthermore, the order of accuracy of the multistage-analytical method can be made higher simply by taking more number of successive iterations in the multistage evolution.

The Richards model and its generalized version are deterministic models that are often implemented to fit and forecast the cumulative number of infective cases in an epidemic outbreak. In this paper we employ a generalized Richards model to predict the cumulative number of COVID-19 cases in Spain and Italy, based on available epidemiological data. To quantify uncertainty in the parameter estimation, we use a parametric bootstrapping approach to construct a 95% confidence interval estimation for the parameter model. Here we assume that the time series data follow a Poisson distribution. It is found that the 95% confidence interval of each parameter becomes narrow with the increasing number of data. All in all, the model predicts daily new cases of COVID-19 reasonably well during calibration periods. However, the model fails to produce good forecasts when the amount of data used for parameter estimations is not sufficient. Based on our parameter estimates, it is found that the early stages of COVID-19 epidemic, both in Spain and in Italy, followed an almost exponentially growth. The epidemic peak in Spain and Italy is respectively on 2 April 2020 and 28 March 2020. The final sizes of cumulative number of COVID-19 cases in Spain and Italy are forecasted to be at 293220 and 237010, respectively.

This paper gives an errata to the paper "New measure of graph irregularity", Electronic Journal of Graph Theory and Applications {\bf 2}(1) (2014), 52-65.

In this paper, several parameters of the non-linear Hirota-Satsuma coupled KdV system were estimated using a hybrid between the Firefly Algorithm (FFA) and the Modified Adomian decomposition method (MADM). It turns out that optimal parameters can significantly improve the solutions when using a suitably selected fitness function for this problem. The results obtained show that the approximate solutions are highly compatible with the exact solutions and that the hybrid method FFA_MADM gives higher efficiency and accuracy compared to the classic MADM method.

In this paper we propose an explicit predictor-corrector finite difference scheme to numerically solve one-dimensional conservation laws with discontinuous flux function appearing in various physical model problems, such as traffic flow and two-phase flow in porous media. The proposed method is based on the second-order MacCormack finite difference scheme and the solution is obtained by correcting first-order schemes. It is shown that the order of convergence is quadratic in the grid spacing for uniform grids when applied to problems with discontinuity. To illustrate some properties of the proposed scheme, numerical results applied to linear as well as non-linear problems are presented.

The susceptible-infected-recovered (SIR) model of the spread of dengue fever for the case of South Sulawesi is considered. Rangkuti’s variational iteration method (RVIM) is recalled. This paper makes four contributions. The first one is to provide a successive approximation method (SAM) for solving the considered model. The second one is to show that SAM and RVIM are identical. Thirdly, a modification of RVIM is proposed. Fourthly, it is shown that the modification leads to an improvement of the accuracy of the method. Both RVIM and the improved version are quite accurate for short time periods. However, the improved version is more accurate and is able to provide more realistic explicit solutions to the model.

In this paper, a new hybrid conjugate gradient method for solving monotone nonlinear equations is introduced. The scheme is a combination of the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP) conjugate gradient methods with the Solodov and Svaiter projection strategy. Using suitable assumptions, the global convergence of the scheme with monotone line search is provided. Lastly, a numerical experiment was used to enumerate the suitability of the proposed scheme for large-scale problems.

For simple graphs G and H the size multipartite Ramsey number mj ( G , H ) is the smallest natural number t such that any arbitrary red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. We studied the size tripartite Ramsey numbers m 3( G , H ) where G=mK1,n and H=P3 . In this paper, we generalize this result. We determine m3(G,H) where G is a star forest, namely a disjoint union of heterogeneous stars, and H=P3 . Moreover, we also determine m2(G,H) for this pair of graphs G and H .

An edge-colored graph G is rainbow k-connected, if there are k-internally disjoint rainbow paths connecting every pair of vertices of G. The rainbow k-connection number of G, denoted by rck(G), is the minimum number of colors needed for which there exists a rainbow k-connected coloring for G. In this paper, we are able to find sharp lower and upper bounds for the rainbow 2-connection number of Cartesian products of arbitrary 2-connected graphs and paths. We also determine the rainbow 2-connection number of the Cartesian products of some graphs, i.e. complete graphs, fans, wheels, and cycles, with paths.

A simple graph G = (V(G),E(G)) admits an H-covering if every edge in E(G) belongs to at least one subgraph of G isomorphic to a given graph H. A total k-labeling φ : V(G) ∪ E(G) → {1,2,..., k} is called to be an H-irregular total k-labeling of the graph G admitting an H-covering if for every two different subgraphs H' and H" isomorphic to H there is wtφ(H') ≠ wtφ(H"), where wtφ(H)= ∑v ∈ V(H) φ(v) + ∑e ∈ E(H) φ(e). The total H-irregularity strength of a graph G, denoted by ths(G,H), is the smallest integer k such that G has an H-irregular total k-labeling. In this paper we determine the exact value of the cycle-irregularity strength of ladders and fan graphs.

Let T be a tree of order n. For any edge labeling f : E → {1,2,3,...} the weight of a path P is the sum of the labels of the edges of P and is denoted by w(P). If the weights of the nC2 paths in T are exactly 1, 2,...,nC2, then f is called a Leech labeling and a tree which admits a Leech labeling is called a Leech tree. In this paper we determine all Leech trees having diameter three. We also prove that the tree obtained from the path Pn =(v1,v2,...,vn) by attaching a pendent vertex at vn-1 is not a Leech tree for all n ≥ 4.

Let G = (V,E) be a simple connected graph with the vertex set V = {1,2,...,n} and sequence of vertex degrees (d1,d2,...,dn) where di denotes the degree of a vertex i ∈ V. With i ∼ j, we denote the adjacency of the vertices i and j in the graph G. The inverse sum indeg (ISI) index of the graph G is defined as ISI(G)=∑i∼j(didj)/(di+dj). Some new upper bounds for the ISI index are obtained in this paper.

Let G be a connected graph with vertex set V(G). The Harary index of a graph is defined as H(G) = ∑u ≠ v 1/d(u, v), where d(u, v) denotes the distance between u and v. In this paper, we determine the extremal graphs with the maximum Harary index among all bipartite graphs of order n with a given matching number, with a given vertex-connectivity and with a given edge-connectivity, respectively.

In this paper, we provide a natural bijection between a special family of block circulant graphs and the graphs of critical pairs of the posets known as generalized crowns. In particular, every graph in this family of block circulant graphs we investigate has a generating block row that follows a symmetric growth pattern of the all ones matrix. The natural bijection provides an upper bound on the chromatic number for this infinite family of graphs.

Hankel matrices are related to a wide range of disparate determinant computations and algorithms and some very attractive computational properties are allocated to them. Also, the Hankel determinants are crucial factors in the research of singularities and power series with integral coefficients. It is specified that the Fekete-Szego functional and the second Hankel determinant are equivalent to H 1 (2) and H 2 (2), respectively. In this study, the upper bounds were obtained for the second Hankel determinant of the subclass of bi-univalent functions, which is defined by subordination. It is worth noticing that the bounds rendered in the present paper generalize and modify some previous results.

A resolving set W is a set of vertices of a graph G(V, E) such that for every pair of distinct vertices u, v ∈ V(G), there exists a vertex w ∈ W satisfying d(u, w) ≠ d(v, w). A resolving set with minimum number of vertices is called metric basis of G. The metric dimension of G, denoted by dim(G), is the minimum cardinality of a resolving set of G. In this paper, we consider (3, 6)-fullerene and (4, 6)-fullerene graphs and compute the metric dimension for these fullerene graphs. We also give conjecture on the metric dimension of (3, 6)-fullerene and (4, 6)-fullerene graphs.

Let G be a subgroup of the symmetric group Sn. Then G has the Erdos-Ko-Rado (EKR) property, if the size of any intersecting subset of G is bounded above by the size of a point stabilizer of G. The aim of this paper is to investigate the EKR and the strict EKR properties of the groups of order pqr, where p, q, r are three prime numbers.

Let G be a graph. A set D ⊆ V(G) is a global dominating set of G if D is a dominating set of G and $\overline G$. γg(G) denotes global domination number of G. A set D ⊆ V(G) is an outer independent global dominating set (OIGDS) of G if D is a global dominating set of G and V(G) − D is an independent set of G. The cardinality of the smallest OIGDS of G, denoted by γgoi(G), is called the outer independent global domination number of G. An outer independent global dominating set of cardinality γgoi(G) is called a γgoi-set of G. In this paper we characterize trees T for which γgoi(T) = γ(T) and trees T for which γgoi(T) = γg(T) and trees T for which γgoi(T) = γoi(T) and the unicyclic graphs G for which γgoi(G) = γ(G), and the unicyclic graphs G for which γgoi(G) = γg(G).

This paper proposes two new classes of ratio estimators for population mean when information on a known auxiliary variable is available in simple random sampling. A combined family of ratio estimators for estimating population mean by combining the two new estimators together in order to minimize the mean square error (MSE) is then suggested. The expressions for the bias and mean square error of all proposed estimators up to the first order of approximation were obtained. The performance of the proposed estimators was compared with that of existing estimators using both a theoretical and a simulation study. The proposed family of estimators was found to be more efficient than the existing estimators.

In this paper, a numerical procedure called multistage optimal homotopy asymptotic method (MOHAM) is introduced to solve multi-pantograph equations with time delay. It was shown that the MOHAM algorithm rapidly provides accurate convergent approximate solutions of the exact solution using only one term. A comparative study between the proposed method, the homotopy perturbation method (HPM) and the Taylor matrix method are presented. The obtained results revealed that the method is of higher accuracy, effective and easy to use.

We correct a small error in a 1996 paper of Albertson and Haas, and extend their lower bound for the fraction of properly colorable edges of planar subcubic graphs that are simple, connected, bridgeless, and edge-maximal to other surface embeddings of subcubic graphs.

The distance signless Laplacian spectral radius of a connected graph G is the largest eigenvalue of the distance signless Laplacian matrix of G‎, ‎defined as ‎D‎Q(G) = Tr(G) + D(G)‎, ‎where D(G) is the distance matrix of G and Tr(G) is the diagonal matrix of vertex transmissions of G‎. ‎In this paper we determine some upper and lower bounds on the distance signless Laplacian spectral radius of G based on its order and independence number‎, ‎and characterize the extremal graph‎. ‎In addition‎, ‎we give an exact description of the distance signless Laplacian spectrum and the distance signless Laplacian energy of the join of regular graphs in terms of their adjacency spectrum‎.

Let G be a graph with an edge k-coloring γ : E(G) → {1, …, k} (not necessarily proper). A path is called a rainbow path if all of its edges have different colors. The map γ is called a rainbow coloring if any two vertices can be connected by a rainbow path. The map γ is called a strong rainbow coloring if any two vertices can be connected by a rainbow geodesic. The smallest k for which there is a rainbow k-coloring (resp. strong rainbow k-coloring) on G is called the rainbow connection number (resp. strong rainbow connection number) of G, denoted rc(G) (resp. src(G)). In this paper we generalize the notion of “color codes” that was originally used by Chartrand et al. in their study of the rc and src of complete bipartite graphs, so that it now applies to any connected graph. Using color codes, we prove a new class of lower bounds depending on the existence of sets with common neighbours. Tight examples are discussed, involving the amalgamation of complete graphs, generalized wheel graphs, and a special class of sequential join of graphs.

For integer k ≥ 2, let X = {0, 1, 2, …, k}. In this paper, we determine the order of a star graph K1, n of n + 1 vertices, such that K1, n admits a topological integer additive set-labeling (TIASL) with respect to a set X. We also give a condition for a star graph K1, n such that K1, n is not a TIASL-graph on set X.

In this paper, we give a generalization of slightly compressible modules. We introduce the notion of M -slightly compressible modules, i.e. a right R module N is called M - slightly compressible if for every nonzero submodule A of N there exists a nonzero R -homomorphism s from M to N such that . Many examples of M -slightly compressible modules are provided. Some results on M -slightly compressible modules are obtained, which are interesting and important.

In this paper, we define the disjunctive sum, difference and Cartesian product of two interval valued neutrosophic sets and study their basic properties. The notions of the (α,β,γ) interval cut set of interval valued neutrosophic sets and the (α,β,γ) strong interval cut set of interval valued neutrosophic sets are put forward. Some related properties have been established with proof, examples and counter examples.

Let A be a nontrivial abelian group. A connected simple graph G = (V, E) is A-antimagic, if there exists an edge labeling f : E(G)→A ∖ {0A} such that the induced vertex labeling f+(v)=∑{u, v}∈E(G)f({u, v}) is a one-to-one map. The integer-antimagic spectrum of a graph G is the set IAM (G)={k : G is ℤk-antimagic and k ≥ 2}. In this paper, we determine the integer-antimagic spectra for all Hamiltonian graphs.

The es-splitting operation on an n-connected binary matroid may not yield an n-connected matroid for (n ≥ 3). In this paper, we show that given an n-connected binary matroid M of rank r, the resulting es-splitting binary matroid has an n-connected minor of rank-(r + 1) having |E(M)| + 1 elements.

A graph G is said to be determined by its Laplacian spectrum (DLS) if every graph with the same Laplacian spectrum is isomorphic to G. A graph which is a collection of hexagons (lengths of these cycles can be different) all sharing precisely one vertex is called a spinner graph. A tree with exactly one vertex of degree greater than 2 is called a starlike tree. If a spinner graph and a starlike tree are joined by merging their vertices of degree greater than 2, then the resulting graph is called a tarantula graph. It is known that spinner graphs and starlike trees are DLS. In this paper, we prove that tarantula graphs are determined by their Laplacian spectrum.

The anti-Ramsey number arn(H) of an r-uniform hypergraph is the maximum number of colors that can be used to color the hyperedges of a complete r-uniform hypergraph on n vertices without producing a rainbow copy of H. In this paper, we determine anti-Ramsey numbers for paths of length 2, certain stars and complete hypergraphs, and the complete 3-uniform hypergraph of order 4 with a single hyperedge removed.

Given a positive integer k and a graph G = (V, E), a function f from V to the power set of Ik is called a k-rainbow function if for each vertex v ∈ V, f(v)=∅ implies ∪u ∈ N(v)f(u)=Ik where N(v) is the set of all neighbors of vertex v and Ik = {1, …, k}. Finding a k-rainbow function of minimum weight of ∑v ∈ V|f(v)|, which is called the k-rainbow domination problem, is known to be NP-complete for arbitrary graphs and values of k. In this paper, we propose a dynamic programming algorithm to solve the k-rainbow domination problem for graphs with bounded tree-width tw in 𝒪((2k + 1+1)twn) time, where G has n vertices. Moreover, we also show that the same approach is applicable to solve the weighted k-rainbow domination problem with the same complexity. Therefore, both problems of k-rainbow and weighted k-rainbow domination belong to the class FPT, or fixed parameter tractable, with respect to tree-width. In addition to formally showing the correctness of our algorithms, we also implemented these algorithms to illustrate some examples.

Given a simple graph G=(V,E) with maximum degree Δ. Let (V0, V1, V2) be an ordered partition of V, where Vi = {v ∈ V : f(v)=i} for i = 0, 1 and V2 = {v ∈ V : f(v)≥2}. A function f : V → {0, 1, …, ⌈Δ/2⌉+1} is a strong Roman dominating function (StRDF) on G, if every v ∈ V0 has a neighbor w ∈ V2 and f(w)≥1 + ⌈1/2|N(w)∩V0|⌉. A function f : V → {0, 1, …, ⌈Δ/2⌉+1} is a unique response strong Roman function (URStRF), if w ∈ V0, then |N(w)∩V2|≤1 and w ∈ V1 ∪ V2 implies that |N(w)∩V2|=0. A function f : V → {0, 1, …, ⌈Δ/2⌉+1} is a unique response strong Roman dominating function (URStRDF) if it is both URStRF and StRDF. The unique response strong Roman domination number of G, denoted by uStR(G), is the minimum weight of a unique response strong Roman dominating function. In this paper we approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, for any tree T of order n ≥ 3 we prove the sharp bound uStR(T)≤8n/9.

A simple graph G=(V(G),E(G)) admits an H-covering if every edge in E(G) belongs at least to one subgraph of G isomorphic to a given graph H. Then the graph G admitting H-covering admits an H-irregular total k-labeling f: V(G) U E(G) \to {1, 2, ..., k} if for every two different subgraphs H' and H'' isomorphic to H there is $wt_{f}(H') \neq wt_{f}(H'')$, where $wt_{f}(H)= \sum \limits_{v\in V(H)} f(v) + \sum \limits_{e \in E(H)} f(e)$ is the associated H-weight. The minimum k for which the graph G has an H-irregular total k-labeling is called the total H-irregularity strength of the graph G.In this paper, we obtain the precise value of the total H-irregularity strength of G-amalgamation of graphs.

The paper discusses bounds on the nullity number of graphs. It is proved in [B. Cheng and B. Liu, On the nullity of graphs. Electron. J. Linear Algebra 16 (2007) 60--67] that $\eta \le n - D$, where $\eta$, n and D denote the nullity number, the order and the diameter of a connected graph, respectively. We first give a necessary condition on the extremal graphs corresponding to that bound, and then we strengthen the bound itself using the maximum clique number. In addition, we prove bounds on the nullity using the number of pendant neighbors in a graph. One of those bounds is an improvement of a known bound involving the domination number.

In decomposition theory, extreme sets have been studied extensively due to its connection to perfect matchings in a graph. In this paper, we first define extreme sets with respect to degree-matchings and next investigate some of their properties. In particular, we prove the generalized Decomposition Theorem and give a characterization for the set of all extreme vertices in a graph.

This paper is a survey on the upper and lower bounds for the largest eigenvalue of the Laplacian matrix, known as the Laplacian spectral radius, of a graph. The bounds are given as functions of graph parameters like the number of vertices, the number of edges, degree sequence, average 2-degrees, diameter, covering number, domination number, independence number and other parameters.

Let $p \ge 3$ be a positive integer and let $k \in {1, 2, ..., p-1} \ \lfloor p/2 \rfloor$. The generalized Petersen graph GP(p,k) has its vertex and edge set as $V(GP(p, k)) = \{u_i : i \in Zp\} \cup \{u_i^\prime : i \in Z_p\}$ and $E(GP(p, k)) = \{u_i u_{i+1} : i \in Z_p\} \cup \{u_i^\prime u_{i+k}^\prime \in Z_p\} \cup \{u_iu_i^\prime : i \in Z_p\}$. In this paper we probe its spectrum and determine the Estrada index, Laplacian Estrada index, signless Laplacian Estrada index, normalized Laplacian Estrada index, and energy of a graph. While obtaining some interesting results, we also provide relevant background and problems.

The distinguishing index D'(G) of a graph G is the least integer d such that G has an edge labeling with d labels that is preserved only by a trivial automorphism. The Kronecker product G x H of two graphs G and H is the graph with vertex set V(G) x V(H) and edge set {{(u,x), (v,y)} |{u,v} ∈ E(G) and {x,y} ∈ E(H)}. In this paper we study the distinguishing index of Kronecker product of two graphs.

This paper discusses the automorphism group of a class of weakly semiregular bipartite graphs and its subclass called WSBEND graphs. It also tries to analyse the automorphism group of the SM sum graphs and SM balancing graphs. These graphs are weakly semiregular bipartite graphs too. The SM sum graphs are particular cases of bipartite Kneser graphs. The bipartite Kneser type graphs are defined on n-sets for a fixed positive integer n. The automorphism groups of the bipartite Kneser type graphs are related to that of weakly semiregular bipartite graphs. Weakly semiregular bipartite graphs in which the neighbourhoods of the vertices in the SD part having the same degree sequence, possess non trivial automorphisms. The automorphism groups of SM sum graphs are isomorphic to the symmetric groups. The relationship between the automorphism groups of SM balancing graphs and symmetric groups are established here. It has been observed by using the well known algorithm Nauty, that the size of automorphism groups of SM balancing graphs are prodigious. Every weakly semiregular bipartite graphs with k-NSD subparts has a matching which saturates the smaller partition.