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A set S of vertices of a connected graph G is a doubly connected dominating set (DCDS) if every vertex not in S is adjacent to some vertex in S and the subgraphs induced by S and V - S are connected. The doubly connected domination number γ cc (G) is the minimum size of such a set. The doubly connected domination subdivision number sd γ cc (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) to increase the doubly connected domination number. It was conjectured (Karami et al. in Mat Vesnic 64:232–239, 2012) that the doubly connected domination subdivision number of a connected planar graph is at most two. In this paper, we disprove this conjecture by showing that the doubly connected domination subdivision number of the regular icosahedron graph is three. [ABSTRACT FROM AUTHOR]

An outer independent (double) Roman dominating function is a (double) Roman dominating function f for which the set of vertices assigned 0 under f is independent. The outer independent (double) Roman domination number ( γ oidR (G) ) γ oiR (G) is the minimum weight taken over all outer independent (double) Roman dominating functions of G. A vertex cover number β (G) is the minimum size of any vertex cover sets of a graph G. In this work, we present some contributions to the study of outer independent double Roman domination in graphs. Characterizations of the families of all connected graphs with small outer independent double Roman domination numbers, and tight lower and upper bounds on this parameter are given. We also prove that the decision problem associated with γ oidR (G) is NP-complete even when restricted to planar graphs with maximum degree at most four. We moreover prove that 2 β (T) + 1 ≤ γ oidR (T) ≤ 3 β (T) for any tree T, and show that each integer between the lower and upper bounds is realizable. Finally, we give an exact formula for this parameter concerning the corona graphs. [ABSTRACT FROM AUTHOR]

A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number is the minimum cardinality of a paired-dominating set of G. The paired-domination subdivision number is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. It was conjectured that the paired-domination subdivision number is at most n - 1 for every connected graph G of order n ≥ 3 which does not contain isolated vertices. In this paper, we settle the conjecture in the affirmative. [ABSTRACT FROM AUTHOR]

A double Roman dominating function (DRDF) on a graph G = (V , E) is a function f : V → { 0 , 1 , 2 , 3 } having the property that if f (v) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f (w) = 3 , and if f (v) = 1 , then vertex v must have at least one neighbor w with f (w) ≥ 2 . The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number γ dR (G) is the minimum weight of a DRDF on G. Recently, Khoeilar et al. (Discrete Appl Math 270:159–167, 2019) proved that a connected graph G of order n with minimum degree two different from C 5 and C 7 , satisfies γ dR (G) ≤ 11 10 n. In this paper, we show that this upper bound can be improved to 20 19 n if G is restricted to connected graphs of order n with minimum degree at least two and no C 5 -cycle such that G ∉ { C 7 , C 11 , C 13 , C 17 }. Moreover, we provide an infinite family of graphs attaining this bound. [ABSTRACT FROM AUTHOR]

For a nondecreasing sequence of integers S = (s 1 , s 2 , ...) an S-packing k-coloring of a graph G is a mapping from V(G) to { 1 , 2 , ... , k } such that vertices with color i have pairwise distance greater than s i . By setting s i = d + ⌊ i - 1 n ⌋ we obtain a (d, n)-packing coloring of a graph G. The smallest integer k for which there exists a (d, n)-packing coloring of G is called the (d, n)-packing chromatic number of G. In the special case when d and n are both equal to one we obtain the packing chromatic number of G. We determine the packing chromatic number of base-3 Sierpiński graphs and provide new results on (d, n)-packing chromatic colorings for this class of graphs. By using a dynamic algorithm, we establish the packing chromatic number for H-graphs. [ABSTRACT FROM AUTHOR]

During the past years, hybrid solar-wind power systems containing photovoltaic (PV) and wind generators are used to minimize the intermittency problem of renewable power generation units. The improved modeling and control schemes for a grid-tied hybrid PV–wind system is presented in the current research work. The maximum power point tracking namely "MPPT" algorithm together with controlling the pitch angle are used, respectively, for the PV system and wind power generation to attain the maximum power for any given external weather conditions. A radial basis function network sliding mode known as the RBFNSM method is used to control the pitch angle in the wind energy system, while the PV system uses a proportional–integral–derivative controller equipped with the fuzzy gain scheduling in order to enhance the transient state and mitigate the settling time to ensure the stability of the mentioned system. To test the suggested control scheme's effectiveness, MATLAB simulations are carried out under various scenarios of the wind speed as well as solar irradiation. The obtained results show the efficiency of the adaptive MPPT method to harness the highest power under very challenging scenarios. The merits of the developed schemes are quickly and precisely tracking the maximum power output of the hybrid PV–wind system. Besides, the power flowing between the utility grid and the hybrid source with a fast transient response and improved stability performance is effectively controlled using the offered schemes. [ABSTRACT FROM AUTHOR]

A molecular network can be characterized by several different ways, like a matrix, a polynomial, a drawing or a topological descriptor. A topological descriptor is a numeric quantity associated with a network or a graph that characterizes its whole structural properties. Analyzing and determining the topological indices and structural properties of a network or a graph have been a worthy studied topic in the field of chemistry, networks analysis, etc. In this paper, we consider several types of the generalized Sierpiński networks and investigate the explicit expressions of some well-known valency-based topological indices. Taking into account the other structural property of the generalized Sierpiński networks, the average degree is determined. [ABSTRACT FROM AUTHOR]

For a graph G, let f : V (G) → P ({ 1 , 2 }). If for each vertex v ∈ V (G) such that f (v) = ∅ we have ⋃ u ∈ N (v) f (u) = { 1 , 2 } , then f is called a 2-rainbow dominating function (2RDF) of G. The weight w(f) of a function f is defined as w (f) = ∑ v ∈ V (G) f (v) . The minimum weight of a 2RDF of G is called the 2-rainbow domination number of G, denoted by γ r 2 (G) . The 2-rainbow domination stability, s t γ r 2 (G) , of G is the minimum number of vertices in G whose removal changes the 2-rainbow domination number. In this paper, we first determine the exact values on 2-rainbow domination stability of some special classes of graphs, such as paths, cycles, complete graphs and complete bipartite graphs. Then we obtain several bounds on s t γ r 2 (G) . In particular, we obtain s t γ r 2 (G) ≤ δ (G) + 1 and s t γ r 2 (G) ≤ | V (G) | - Δ (G) - 1 if γ r 2 (G) ≥ 3 . Moreover, we prove that there exists no graph G with s t γ r 2 (G) = | V (G) | - 2 when n ≥ 4 and characterize the graphs G with s t γ r 2 (G) = | V (G) | - 1 or s t γ r 2 (G) = | V (G) | - 3 . [ABSTRACT FROM AUTHOR]

Based on the indefinite error-monitoring function, we propose a novel Zhang neural network (ZNN) model called NNT-ZNN with two properties of nonlinear and noise-tolerant for the time-varying and static matrix square root finding in this paper. Compared to the existing models associated with the square matrix root finding, the NNT-ZNN model proposed in this study fully takes error caused by possible noise on ZNN hardware implementation into account. Under the background that the large model-implementation error, the model still has the ability to converge to the theoretical square root of the given matrix with simulative results illustrated in the paper. For the purpose of comparison, the ZNN model proposed by Zhang et al. is also introduced. Beyond that, the corresponding convergence results of the NNT-ZNN model corresponding to various activation functions, are also shown via time-varying and static positive definite matrix. In the end, the experiments are simulated with MATLAB, which further verifies the availability, effectiveness of the proposed NNT-ZNN model, and robustness against unknown noise. [ABSTRACT FROM AUTHOR]

For a graph G, let f : V (G) → P ({ 1 , 2 }). If for each vertex v ∈ V (G) such that f (v) = ∅ we have ⋃ u ∈ N (v) f (u) = { 1 , 2 } , then f is called a 2-rainbow dominating function (2RDF) of G. The weightw(f) of f is defined as w (f) = ∑ v ∈ V (G) f (v) . The minimum weight of a 2RDF of G is called the 2-rainbow domination number of G, which is denoted by γ r 2 (G) . A graph G is 2-rainbow domination stable if the 2-rainbow domination number of G remains unchanged under removal of any vertex. In this paper, we prove that determining whether a graph is 2-rainbow domination stable is NP-hard and characterize 2-rainbow domination stable trees. [ABSTRACT FROM AUTHOR]

Given a positive integer t and a graph F, the goal is to assign a subset of the color set { 1 , 2 , ... , t } to every vertex of F such that every vertex with the empty set assigned has all t colors in its neighborhood. Such an assignment is called the t-rainbow dominating function ( t RDF ) of the graph F. A t RDF is independent ( I t RDF ) if vertices assigned with non-empty sets are pairwise non-adjacent. The weight of a t RDF g of a graph F is the value w (g) = ∑ v ∈ V (F) | g (v) | . The independent t-rainbow domination number i r t (F) is the minimum weight over all I t RDF s of F. In this article, it is proved that the independent t-rainbow domination problem is NP-complete even if the input graph is restricted to a bipartite graph or a planar graph, and the results of the study provide some bounds for the independent t-rainbow domination number of any graph for a positive integer t. Moreover, the exact values and bounds of the independent t-rainbow domination numbers of some Petersen graphs and torus graphs are given. [ABSTRACT FROM AUTHOR]

In this study, the potential of boron nitride (B21N21), aluminum phosphide (Al21P21), carbon (C24) and silicon (Si24) nanocages as anode electrodes of Lithium-ion (Li-ion), Sodium-ion (Na-ion) and Potassium-ion (K-ion) batteries has been investigated. The effects of halogen adoption of studied nanocages on ability of metal-ion batteries have been examined. Results show that the Al21P21 nanocage as anode electrode in metal-ion batteries has higher potential than B21N21, C24 and Si24 nanocages. It's found that the K-ion battery has higher cell voltage and higher performance than Li-ion and Na-ion batteries; the halogen adoption of studied nanocages increases the cell voltage of metal-ion batteries; the Fluorine F-doped metal-ion batteries have higher cell voltage than Chlorine Cl- and Bromine Br-doped metal-ion batteries. Finally, the F-doped Al20P21 was proposed as novel anode electrode in K-ion battery with the highest performance. [ABSTRACT FROM AUTHOR]

A set S⊆V(G) is a k-distance independent set of a graph G if the distance between every two vertices of S is greater than k. The k-distance independence number αk(G) of G is the maximum cardinality over all k-distance independent sets in G. A k-distance coloring of G is a function f from V(G) onto a set of positive integers (colors) such that for any two distinct vertices u and v at distance less than or equal to k we have f(u)≠f(v). The k-distance chromatic number of a graph G is the smallest number of colors needed to have a k-distance coloring of G. The k-distance independence numbers and 2-distance chromatic numbers of Cartesian products of cycles are considered. A computer-aided method with the isomorphic rejection is used to determine the k-distance independence numbers of Cartesian products of cycles. By using these results, several lower and upper bounds on the maximal cardinality AqL(n,d) of a q-ary Lee code of length n with a minimum distance at least d are improved. It is also established that the 2-distance chromatic number of G equals |V(G)|α(G2) for G=Cm□Cn□Ck, whenever k≥3 and (m,n)∈{(3,3),(3,4),(3,5),(4,4)} or k, m and n are all multiples of seven. Moreover, it is shown that the 2-distance chromatic number of the three-dimensional square lattice is equal to seven. [ABSTRACT FROM AUTHOR]

In this paper we obtain some bounds and exact values of the total domination numbers and open packing numbers for pyrene torus, hexabenzocoronene, H-phenylenic nanotube and H-napthelenic nanotube. [ABSTRACT FROM AUTHOR]

For a graph $$G=(V, E)$$ , a weak $$\{2\}$$ -dominating function $$f:V\rightarrow \{0,1,2\}$$ has the property that $$\sum _{u\in N(v)}f(u)\ge 2$$ for every vertex $$v\in V$$ with $$f(v)= 0$$ , where N( v) is the set of neighbors of v in G. The weight of a weak $$\{2\}$$ -dominating function f is the sum $$\sum _{v\in V}f(v)$$ and the minimum weight of a weak $$\{2\}$$ -dominating function is the weak $$\{2\}$$ -domination number. In this paper, we introduce a discharging approach and provide a short proof for the lower bound of the weak $$\{2\}$$ -domination number of $$C_n \Box C_5$$ , which was obtained by Stȩpień, et al. (Discrete Appl Math 170:113-116, 2014). Moreover, we obtain the weak $$\{2\}$$ -domination numbers of $$C_n \Box C_3$$ and $$C_n \Box C_4$$ . [ABSTRACT FROM AUTHOR]

Let $$G=(V, E)$$ be a graph. Denote $$d_G(u, v)$$ the distance between two vertices u and v in G. An L(2, 1)-labeling of G is a function $$f: V \rightarrow \{0,1,\ldots \}$$ such that for any two vertices u and v, $$|f(u)-f(v)| \ge 2$$ if $$d_G(u, v) = 1$$ and $$|f(u)-f(v)| \ge 1$$ if $$d_G(u, v) = 2$$ . The span of f is the difference between the largest and the smallest number in f( V). The $$\lambda $$ -number $$\lambda (G)$$ of G is the minimum span over all L(2, 1)-labelings of G. In this paper, we conclude that the $$\lambda $$ -number of each brick product graph is 5 or 6, which confirms Conjecture 6.1 stated in Li et al. (J Comb Optim 25:716-736, 2013). [ABSTRACT FROM AUTHOR]

The semitotal domination number of a graph G without isolated vertices is the minimum cardinality of a set S of vertices of G such that every vertex in $$V(G){\setminus } S$$ is adjacent to at least one vertex in S, and every vertex in S is within distance 2 of another vertex of S. In Henning and Marcon (Ann Comb 20(4):1-15, 2016), it was shown that every connected claw-free cubic graph G of order n has semitotal domination number at most $$\frac{4n}{11}$$ when $$n\ge 10$$ , and it was conjectured that the bound can be improved from $$\frac{4n}{11}$$ to $$\frac{n}{3}$$ if $$G\notin \{K_4,N_2\}$$ . In this paper, we prove this conjecture. [ABSTRACT FROM AUTHOR]

The frequency assignment problem (FAP) asks for assigning frequencies (channels) in a wireless network from the available radio spectrum to the transceivers of the network. One of the graph theoretical models of FAP is the L(3, 2, 1)-labeling of a graph, which is an abstraction of assigning integer frequencies to radio transceivers such that (i) transceivers that are one unit of distance apart receive frequencies that differ by at least three, (ii) transceivers that are two units of distance apart receive frequencies that differ by at least two, and (iii) transceivers that are three units of distance apart receive frequencies that differ by at least one. The relaxation of the L(3, 2, 1)-labeling called the ( s, t, r)-relaxed k- L(3, 2, 1)-labeling is proposed in this paper. This concept is a generalization of the ( s, t)-relaxed k- L(2, 1)-labeling (Lin in J Comb Optim 2016, doi:). Basic properties of ( s, t, r)-relaxed k- L(3, 2, 1)-labeling are discussed and optimal ( s, t, r)-relaxed k- L(3, 2, 1)-labelings for paths and some cycles as well as for the hexagonal lattice and the square lattice are determined. [ABSTRACT FROM AUTHOR]

Let $$G=(V, E)$$ be a graph. Denote $$d_G(u, v)$$ the distance between two vertices $$u$$ and $$v$$ in $$G$$ . An $$L(2, 1)$$ -labeling of $$G$$ is a function $$f: V \rightarrow \{0,1,\cdots \}$$ such that for any two vertices $$u$$ and $$v$$ , $$|f(u)-f(v)| \ge 2$$ if $$d_G(u, v) = 1$$ and $$|f(u)-f(v)| \ge 1$$ if $$d_G(u, v) = 2$$ . The span of $$f$$ is the difference between the largest and the smallest number in $$f(V)$$ . The $$\lambda $$ -number of $$G$$ , denoted $$\lambda (G)$$ , is the minimum span over all $$L(2,1 )$$ -labelings of $$G$$ . In this article, we confirm Conjecture 6.1 stated in X. Li et al. (J Comb Optim 25:716-736, ) in the case when (i) $$\ell $$ is even, or (ii) $$\ell \ge 5$$ is odd and $$0 \le r \le 8$$ . [ABSTRACT FROM AUTHOR]

An acyclic $$k$$ -coloring of a graph $$G$$ is a $$k$$ -coloring of its vertices such that no cycle of $$G$$ is bichromatic. $$G$$ is called acyclically $$k$$ -colorable if it admits an acyclic $$k$$ -coloring. In this paper, we prove that the generalized Petersen graph $$P(n,k)$$ is acyclically 3-colorable except $$P(4,1)$$ and the classical Petersen graph $$P(5,2)$$ . [ABSTRACT FROM AUTHOR]

The $$L(3,2,1)$$ -labeling of a graph is an abstraction of assigning integer frequencies to radio transceivers such that (1) transceivers that are one unit of distance apart receive frequencies that differ by at least three, (2) transceivers that are two units of distance apart receive frequencies that differ by at least two, and (3) transceivers that are three units of distance apart receive frequencies that differ by at least one. The least span of frequencies in such a labeling is referred to as the $$\uplambda _{3,2,1}$$ -number of the graph. In this paper, we determine the $$\uplambda _{3,2,1}$$ -number for toroidal grids $$T_{n,m}, m \le 6$$ and the $$\uplambda _{3,2,1}$$ -number for the triangular grid. The last result proves the conjecture from Calamoneri (Inf Process Lett 113:361-364, ). [ABSTRACT FROM AUTHOR]

In this note, a dynamic programming-like method is used to detect $$k$$-term semi-progression efficiently. By using this approach, we obtain some exact values and new lower bounds on semi-progression van der Waerden numbers. [ABSTRACT FROM AUTHOR]