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HD 69479/80 is a composite‐spectrum binary whose components are a late‐G giant and an early‐A dwarf. The orbit has a period of only 91 days (which seems short for a system containing a cool giant with a radius of ∼13 solar radii), and a very small, but probably significantly non‐zero, eccentricity. We separated the component spectra by a procedure of spectral subtraction, using a standard single giant spectrum as a template, and found the closest match to the spectrum of the cool component to be that of 15 Cyg (G8 III). We measured the radial velocity of the secondary component from each uncovered spectrum, solved the SB2 orbit for the system, and derived a mass ratio m1/m2 of 1.318. Fitting synthetic spectra to the spectra of the secondary component indicated a Teff of 9250 K, log g = 3.75, and a rotational velocity of ∼90 km s−1. We determined the difference in absolute magnitude, δV, between the component stars to be 1.07 mag, the late‐type component being the brighter; we could thence calculate radii and luminosities for both components, plot their H–R diagram positions, and fit evolutionary tracks. The best‐fitting tracks indicated masses of 2.9 M⊙ for the giant and 2.2 M⊙ for the dwarf, which was fully in keeping with the mass ratio given by the SB2 orbit. The track for the dwarf star confirms that this component has begun to evolve away from the ZAMS. Fitting the corresponding isochrone to those H–R diagram positions indicated a log (age) of the system of approximately 8.60 Gyr since the cool star evolved from the ZAMS, which is a little younger than the ages deduced for many cool giants. We also detected the λ 6707‐Å lithium line in the spectrum of the giant component, thus adding to the evidence that it is near the start of its primary ascent of the red‐giant branch. [ABSTRACT FROM AUTHOR]

The article discusses the most-cited researches in history. Topics include topics of papers and citations that has allowed them to reach the top the most cited research like Bioinformatics, Phylogenetics and Statistics, a chart showing the top 10 most cited researches and information on the first systematic effort to track citations the Science Citation Index (SCI).

For a graph G, we define a total k-labeling ϕ as a combination of an edge labeling ϕe : E(G) → {1, 2, . . ., ke} and a vertex labeling ϕv : V (G) → {0, 2, . . ., 2kv}, where k = max {ke, 2kv}. The total k-labeling ϕ is called a vertex irregular reflexive k-labeling of G if any pair of vertices u, u' have distinct vertex weights wtϕ(u) ≠ wtϕ(u'), where wtϕ(u) = ϕ(u) + Σuu'∈E(G) ϕ(uu' ) for any vertex u ∈ V (G). The smallest value of k for which such a labeling exists is called the reflexive vertex strength of G, denoted by rvs(G). In this paper, we present a new lower bound for the reflexive vertex strength of any graph. We investigate the exact values of the reflexive vertex strength of generalized friendship graphs, corona product of two paths, and corona product of a cycle with isolated vertices by referring to the lower bound. This study discovers some interesting open problems that are worth further exploration. [ABSTRACT FROM AUTHOR]

Several graphs are presented which provides supplementary information to the paper "Ten Years of MIPAS measurements with ESA Level 2 processor V6-Part I: retrieval algorithm and diagnostics of the products," by P. Raspollini and colleagues including one on the Ozone percent random and systematic error profiles, another on the Ozone Averaging Kernels (AKs), and one on the temperature AKs.

The author reflects on the graphic form x-y scatterplot which is very useful to science. It states that truth was the first rule of the graph and the design, helps to achieve the goal and allows the viewer to show the important trends and explain them the theory. It also mentions that the graphs can help in data exploration and data presentation.

A chart is presented that lists highly cited metabolic syndrome-related research papers for 2003-2005.

The concept of topology defined on a set can be extended to the field of graph theory by defining the notion of graph topologies on graphs where we consider a collection of subgraphs of a graph G in such a way that this collection satisfies the three conditions stated similarly to that of the three axioms of point-set topology. This paper discusses an introduction and basic concepts to the graph topology. A subgraph of G is said to be open if it is in the graph topology TG. The paper also introduces the concept of the closed graph and the closure of graph topology in graph topological space using the ideas of decomposition-complement and neighborhood-complement. [ABSTRACT FROM AUTHOR]

The article discusses modern medical case histories published between 1600 and 1900, focusing on the use of seriality by the hospital system and its use of tables. The condensing of narrative patient information into tabular form allowed for the creation of quantitive data reporting which would support rhetorical arguments. Other subjects considered include the impact of the history of printing technologies on the history of science, the development of scientific specialization, and the creation of indexes to access knowledge and information in books.

This paper investigates the Weierstrass type function in local fields whose graph is a chaotic repelling set of a discrete dynamical system, and proves that their exists a linear connection between the orders of its p-adic calculus and the dimensions of the corresponding graphs. [ABSTRACT FROM AUTHOR]

The article focuses on SYMOZ, a two-player mathematical game played in 9-by-9 size graph paper. It reveals the three kinds of mesh points that should be considered in placing the ships in the graph as well as the basic rules that must be satisfied in placing the ship. It notes on the significance of declaring the number of each mesh points in determining the location of the ship. Figures showing the procedure and solutions of the game are also presented.

Presents a series of diagrams showing DNA at different scales. Use of the diagrams in teaching about genetic material.

Several tables are presented that list reseach papers related to cancer which gain the highest number of citations including "Sorafenib in advanced hepatocellular carcinoma," by J. M. Llovet and colleagues in 2008, "Gefitinib or carboplatin-paclitaxel in pulmonary adenocarcinoma," by T. S. Mok in 2009 and "A comprehensive catalogue of somatic mutations from a human cancer genome," by E. D. Pleasance in 2010.

When applied to Tai varieties beyond Southwestern Tai, Gedney's tone box framework and its revised versions have several shortcomings, the most notable of which is the lack of consideration of diachronic issues that give rise to more complicated tonal correspondences. This paper proposes a more widely applicable Tai tone box chart with one additional tier based on the more probable tone-conditioning phonetic nature of onsets found in non-Southwestern Tai varieties. This Tai tone box chart has been designed to be paired with an ancillary chart to explain the essential diachronic processes of different Tai varieties. It thus provides insights into the revision of Proto-Tai phonological reconstructions based on tonology and support for the early-stage Southern-Northern Tai division and the Southwestern-Central Tai versus Yongnan Zhuang-Saek-Northern Zhuang divisions. [ABSTRACT FROM AUTHOR]

The energy of a graph G, denoted by E(G), is defined as the sum of absolute values of all eigenvalues of G. A graph G is called reciprocal if 1/≥ is an eigenvalue of G whenever λ is an eigenvalue of G. Further, if λ and 1 λ have the same multiplicities, for each eigenvalue λ, then it is called strong reciprocal. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631{633), it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree ∞(G) whose adjacency matrix is non-singular, E(G) λ Δ(G) + ∞(G) and the equality holds if and only if G is a complete graph. Here, we prove the validity of this conjecture for some strong reciprocal graphs. Moreover, we show that if G is a strong reciprocal graph, then E(G) λ Δ(G) + ∞(G) - 1 2. Recently, it has been proved that if G is a reciprocal graph of order n and its spectral radius, ?, is at least 4λmin, where λmin is the smallest absolute value of eigenvalues of G, then E(G) λ n + 1 2. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption. [ABSTRACT FROM AUTHOR]

This paper presents and studies a generalization of the microscopic image reconstruction problem (MIR) introduced by Frosini and Nivat (2007) and Nivat (2002). Consider a specimen for inspection, represented as a collection of points typically organized on a grid in the plane. Assume each point x has an associated physical value ℓ x , which we would like to determine. However, it might be that obtaining these values precisely (by what we call a surgical probe) is difficult, risky, or impossible. The alternative is to employ aggregate measuring techniques (such as EM, CT, US or MRI), whereby each measurement is taken over a larger window, and the exact values at each point are subsequently extracted by computational methods. In this paper, we extend the MIR framework in a number of ways. First, we consider a generalized setting where the inspected object is represented by an arbitrary graph G , and the vector ℓ ∈ R n assigns a value ℓ v to each node v. A probe centred at a node v will capture a window encompassing its entire neighbourhood N [ v ] , i.e., the outcome of a probe centred at v is P v = ∑ w ∈ N [ v ] ℓ w . We give a criterion for the graphs for which the extended MIR problem can be solved by extracting the vector ℓ from the collection of probes, P = { P v ∣ v ∈ V }. We then consider cases where such reconstruction is impossible (namely, graphs G for which the probe vector P is inconclusive, in the sense that there may be more than one vector ℓ yielding P). Assume that surgical probes are technically available, yet are expensive or risky, and must be used sparingly. We show that in such cases, it may still be possible to achieve reconstruction based on a combination of a collection of ordinary (aggregate) probes together with a suitable set of surgical probes. We aim at identifying the minimum number of surgical probes necessary for a unique reconstruction, depending on the graph topology. This is referred to as the Minimum Surgical Probing problem (MSP). Besides providing a solution for the above problems for arbitrary graphs, we also explore the range of possible behaviours of the Minimum Surgical Probing problem by determining the number of surgical probes necessary in certain specific graph families, such as perfect k -ary trees, paths, cycles, grids, tori, tubes and hypercubes. [ABSTRACT FROM AUTHOR]

The branching index (also known as the connectivity index), introduced in Milan Randić's seminal paper (J Am Chem Soc 97(23):6609–6615, 1975), is one of the the most famous, investigated, and applied among the graph-theoretical molecular descriptors. The atom-bond connectivity (ABC) index (Estrada et al in Indian J Chem A 37:849–855, 1998) and the sum-connectivity (SC) index (Zhou and Trinajstić in J Math Chem 46:1252–1270, 2009) belong to the class of successful variants of the connectivity index. In the present paper, by amalgamating the core idea of the SC and ABC indices, a new molecular descriptor is put forward—the atom-bond sum-connectivity (ABS) index. The graphs attaining the extreme values of the ABS index are determined over the classes of (molecular) trees and general graphs of a fixed order. A noteworthy property of the ABC index is that it increases when a non-isolated edge is inserted between any two non-adjacent vertices. It is proved that this property holds also for the ABS index. [ABSTRACT FROM AUTHOR]

In a graph G , a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S. The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G. In Sedlar and Škrekovski (0000) we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of these configurations is present in the graph. In this paper we extend the result to cactus graphs i.e. graphs in which all cycles are pairwise edge disjoint. We do so by defining a unicyclic subgraph of G for every cycle of G and applying the already introduced approach for unicyclic graphs which involves the configurations. The obtained results enable us to prove the cycle rank conjecture for cacti. They also yield a simple upper bound on metric dimensions of cactus graphs and we conclude the paper by conjecturing that the same upper bound holds in general. [ABSTRACT FROM AUTHOR]

A numbering f of a graph G is a labeling that assigns distinct elements of the set 1 , 2 , ... , V G to the vertices of G. The strength str f G of a numbering f : V G → 1 , 2 , ... , V G of G is defined by str f G = max f u + f v u v ∈ E G , and the strength str G of a graph G is str G = min str f G f is a numbering of G . In this paper, we investigate minimum degree conditions for the strength of graphs. From the study conducted in this paper, certain degree sequences of graphs naturally arise and we provide a proof of the fact that these degree sequences determine a unique graph realization. In addition, we establish a parallel bandwidth result to the one on strength of graphs. Furthermore, we enlarge the class of k -stable properties known so far. These results follow naturally from the work conducted in this paper. [ABSTRACT FROM AUTHOR]

• C -probe graphs are graphs having stable sets N for which edges can be added to obtain a supergraph in C. • A graph G is an (r , ℓ) -graph if V (G) can be partitioned into r independent sets and ℓ cliques. • A graph G is well-covered if every maximal independent set is also maximum. • This paper studies the complexity of recognizing probe graphs for the class of (r , ℓ) -well-covered graphs. Let C be a class of graphs. A graph G = (V , E) is C -probe if V (G) can be partitioned into two sets: non-probes N and probes P , where N is an independent set and new edges may be added between some non-probe vertices such that the resulting graph is in the class C. In this case, we say that (N , P) is a C - probe partition of G. In the Unpartitioned Probe problem for a graph class C we are given a graph G and asked whether G has a C -probe partition, i.e., such a problem consist of recognizing the class of C -probe graphs. A graph G = (V , E) is an (r , ℓ) -graph when V can be partitioned into (S 1 , S 2 , ... , S r , K 1 , K 2 , ... , K ℓ) such that S 1 , S 2 , ... , S r are independent sets, and K 1 , K 2 , ... , K ℓ are cliques. A graph G is well-covered if every maximal independent set is also maximum, and it is (r , ℓ) -well-covered if it is well-covered as well as an (r , ℓ) -graph. In this paper, we study the complexity of the Unpartitioned Probe problem for the class of (r , ℓ) -well-covered graphs. We classify all but the (2 , 0) and (1 , 2) cases. [ABSTRACT FROM AUTHOR]

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. Recently, the crossing numbers of join products of two graphs have been studied. In the paper, we extend know results concerning crossing numbers of join products of small graphs with discrete graphs. The crossing number of the join product G* + Dn for the disconnected graph G* consisting of five vertices and of three edges incident with the same vertex is given. Up to now, the crossing numbers of G + Dn were done only for connected graphs G. In the paper also the crossing numbers of G* + Pn and G* + Cn are given. The paper concludes by giving the crossing numbers of the graphs H + Dn, H + Pn, and H + Cn for four different graphs H with |E(H)| ≤ |V(H)|. The methods used in the paper are new. They are based on combinatorial properties of cyclic permutations. [ABSTRACT FROM AUTHOR]

Let G be a nontrivial edge-colored connected graph. An edge-cut R of G is called a rainbow-cut if no two of its edges are colored the same. An edge-colored graph G is rainbow disconnected if for every two vertices u and v of G, there exists a u-v-rainbow-cut separating them. For a connected graph G, the rainbow disconnection number of G, denoted by rd(G), is defined as the smallest number of colors that are needed in order to make G rainbow disconnected. In this paper, we first determine the maximum size of a connected graph G of order n with rd(G) = k for any given integers k and n with 1 ≤ k ≤ n-1, which solves a conjecture posed only for n odd in [G. Chartrand, S. Devereaux, T.W. Haynes, S.T. Hedetniemi and P. Zhang, Rainbow disconnection in graphs, Discuss. Math. Graph Theory 38 (2018) 1007-1021]. From this result and a result in their paper, we obtain Erdős-Gallai type results for rd(G). Secondly, we discuss bounds on rd(G) for complete multipartite graphs, critical graphs with respect to the chromatic number, minimal graphs with respect to the chromatic index, and regular graphs, and we also give the values of rd(G) for several special graphs. Thirdly, we get Nordhaus-Gaddum type bounds for rd(G), and examples are given to show that the upper and lower bounds are sharp. Finally, we show that for a connected graph G, to compute rd(G) is NP-hard. In particular, we show that it is already NP-complete to decide if rd(G) = 3 for a connected cubic graph. Moreover, we show that for a given edge-colored (with an unbounded number of colors) connected graph G it is NP-complete to decide whether G is rainbow disconnected. [ABSTRACT FROM AUTHOR]

Let I = (V , L) be a vertex-line incidence system which is said to be thin or thick if every line contains two or three vertices, respectively. In either case any two lines can intersect in at most one vertex. Let Γ be the collinearity graph of I and let G be a group of automorphisms of I. Then Γ is said to be locally projective in dimension n with respect to G if (a) G is flag-transitive on I ; (b) the stabilizer G (x) of a vertex x ∈ V in G induces on the set of lines containing x , the group L n (2) for some n ≥ 2 in its natural doubly transitive action of degree 2 n − 1. Thus a locally projective graph in dimension n has valency 2 n − 1 or 2 (2 n − 1) depending on whether it is thin or thick. The G (x) -invariant structure of a projective G F (2) -space of linear dimension n on the set of lines containing x , will be denoted by Π x. In the thin case we further assume that (c) the action of G on Γ is 2- but not 3-arc transitive and (d) the action is of collineation type, meaning that for an edge { x , y } of Γ an element in G which swaps x and y can be chosen to centralize G (x) ∩ G (y) modulo O 2 (G (x) ∩ G (y)). The first examples of locally projective graphs come from affine (thin) and projective (thick) G F (2) -spaces. Less trivial classical examples are the dual polar G F (2) -spaces of orthogonal (thin) and symplectic (thick) types. Further examples coming from Petersen and Tilde geometries are associated with some sporadic simple groups. The fundamental problem is to establish an upper bound on | G (x) | in terms of the dimension n , by finding an absolute bound d such that the vertex-wise stabilizer in G of the ball of radius d in Γ centred at a vertex x is always trivial (independently of the graph): G d (x) = 1. In the classical examples G 2 (x) = 1. The first examples of Petersen and Tilde geometries were constructed in the early 1980s to disprove the belief that the classical bound holds in general. In fact, for the vertex-transitive graph of the Petersen geometry of the Baby Monster group the smallest d with G d (x) = 1 is 5. In 1990s V.I. Trofimov in a brilliant series of papers proved that G 6 (x) = 1 in the thin case without assuming (c) and (d). In 2002 it has been proved by S.V. Shpectorov and the author that in the thin case under consideration all the examples with the critical parameter d above the classical bound indeed come from Petersen geometries (one way or another). An ongoing project is to extend the classification of locally projective amalgams to the thick case, which is considerably more complicated. Until recently only a characterization of an important class of 3-dimensional locally projective graphs [3] was available. In a recent paper [6] the author has shown that if the standard geometrization of a thick locally projective graph contains planes locally isomorphic to the generalized quadrangle of order (2 , 2) , then Γ contains a family of thin locally projective subgraphs which are densely embedded in Γ in the sense that the stabilizer in G (x) of such a subgraph containing x has index 2 n. This result places in a general setting the famous embeddings G (M 22) ↪ G (M 24) , G (C o 2) ↪ G (C o 1) , G (B M) ↪ G (M) among Petersen and Tilde geometries. In the present paper we characterize the graphs of the geometries on the right hand side of these embeddings in the class of thick locally projective graphs with planes of the above specified type and with critical parameter d above the classical bound. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple, connected graph of order n and size m. Then, G = (V , E) is said to be edge (m , k) -choosable, if there exists a collection of subsets of the edge set, { S k (e) : e ∈ E (G) } of cardinality k , such that S k (e 1) ∩ S k (e 2) = ∅ , whenever e 1 and e 2 are incident. This paper initiates a study on edge (m , k) -choosability of certain fundamental classes of graphs and determines the maximum value of k , for which the given graph G is edge (m , k) -choosable. Also, in this paper, the relation between edge choice number and other graph theoretic parameters is discussed and we have given a conjecture on the relation between edge choice number and matching number of a graph. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) , a subset D of vertex set V , is a dominating set of G if every vertex not in D is adjacent to at least one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set) , if the subgraph induced by D in G has a perfect matching. The Min-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the Min-PD problem is NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the other side, the problem is efficiently solvable for many graph classes including interval graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graphs. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NP-complete for planar graphs, which answers an open question asked by Lin et al. (2015) [19] and (2020) [20]. Alvarado et al. (2015) [1] conjectured that given a graph G = (V , E) , it is NP-hard to decide whether γ (G) = γ p r (G). In this paper we settle this conjecture affirmatively. [ABSTRACT FROM AUTHOR]

Let G be a simple graph and A(G) be its adjacency matrix. We say that the graph G satisfies reciprocal eigenvalue property (or simply, property (R)) if 1 λ is an eigenvalue of A(G), whenever λ is an eigenvalue of A(G). The graph G satisfies strong anti-reciprocal property (or simply, property (-SR)) if - 1 λ is an eigenvalue of A(G) with multiplicity k, whenever λ is an eigenvalue of A(G) with multiplicity k. Recently, Barik, Mondal and Pati proved that there exists no singular (non-trivial) tree whose nonzero eigenvalues satisfy reciprocal eigenvalue property. As a concluding remark, the authors mentioned that "It is interesting to see that our result (for trees) is true for any connected graph". In this paper, we prove that there exists no singular unicyclic graph whose nonzero eigenvalues satisfy reciprocal eigenvalue property. Ahmad, Hameed and Jabeen in their recent paper raised the question about existence of noncorona graphs with property (-SR) and constructed seven classes of graphs satisfying this property. Here, we give several families of noncorona graphs with property (-SR). [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a simple graph with vertex set V and edge set E. An {outer-independent total 2 -rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1,2} such that the following conditions hold: (i) for any vertex v with f(v)=∅ we have ⋃u∈NG(v)f(u)={1,2}, (ii) the set of all vertices v∈V(G) with f(v)=∅ is independent and (iii) {v∣f(v)≠∅} has no isolated vertex. The outer-independent total 2 -rainbow domination number of G, denoted by γoitr2(G), is the minimum value of ω(f)=∑v∈V(G)|f(v)| over all such functions f. In this paper, we study the outer-independent total 2 -rainbow domination number of G and classify all graphs with outer-independent total 2 -ainbow domination number belonging to the set {2,3,n}. Among other results, we present some sharp bounds concerning the invariant. [ABSTRACT FROM AUTHOR]

For a (molecular) graph, the second Zagreb index M2(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices. Roman dominating function RDF of G is a function f:V(G)→{0,1,2} satisfying the condition that every vertex with label 0 is adjacent to a vertex with label 2. The weight of an RDF f is w(f)=∑v∈V(G)f(v). The Roman domination number of G, denoted by γR(G), is the minimum weight among all RDF in G. In this paper, we present a lower bound on the second Zagreb index of trees with n vertices and Roman domination number and thus settle one problem given in [On the Zagreb indices of graphs with given Roman domination number, Commun. Comb. Optim. DOI: 10.22049/CCO.2021.27439.1263 (article in press)]. [ABSTRACT FROM AUTHOR]

In this paper, we have punctured unit Zq -Simplex code and constructed a new code called unit Zq -Simplex code of type α. In particular, we find the parameters of these codes and have proved that it is an [ϕ(q)+2, 2, ϕ(q)+2-ϕ(q)ϕ(p)] Zq -linear code if k=2 and [ϕ(q)k-1ϕ(q)-1+ϕ(q)k-2, k, ϕ(q)k-1ϕ(q)-1+ϕ(q)k-2-(ϕ(q)ϕ(p))(ϕ(q)k-1-1ϕ(q)-1+ϕ(q)k-3)] Zq -linear code if k≥3, where p is the smallest prime divisor of q. For q is a prime power and rank k=3, we have given the weight distribution of unit Zq -Simplex codes of type α. Also, we have introduced some new code from Zq -Simplex code called zero divisor Zq -Simplex code and proved that it is an [ρk-1ρ-1,k,ρk-1ρ-1-(ρ(k-1)-1ρ-1)(qp)] Zq -linear code, where ρ=q-ϕ(q) and p is the smallest prime divisor of q. Further, we obtain weight distribution of zero divisor Zq -Simplex code for rank k=3 and q is a prime power. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A coalition in a graph G=(V,E) consists of two disjoint sets V1 and V2 of vertices, such that neither V1 nor V2 is a dominating set, but the union V1∪V2 is a dominating set of G. A coalition partition in a graph G of order n=|V| is a vertex partition π=V1,V2,...,Vk such that every set Vi either is a dominating set consisting of a single vertex of degree n-1, or is not a dominating set but forms a coalition with another set Vj. Associated with every coalition partition π of a graph G is a graph called the coalition graph of G with respect to π, denoted CG(G,π), the vertices of which correspond one-to-one with the sets V1,V2,...,Vk of π and two vertices are adjacent in CG(G,π) if and only if their corresponding sets in π form a coalition. In this paper, we initiate the study of coalition graphs and we show that every graph is a coalition graph. [ABSTRACT FROM AUTHOR]

Let S=(G,σ) be a signed graph. A function f:V→{0,1,2} is a Roman dominating function on S if (i) for each v € V, f(N[v])=f(v)+∑u€N(v)σ(uv)f(u)≥1 and (ii) for each vertex v with f(v)=0 there exists a vertex u €N+(v) such that f(u)=2. In this paper we initiate a study on Roman dominating function on signed graphs. We characterise the signed paths, cycles and stars that admit a Roman dominating function. [ABSTRACT FROM AUTHOR]

Let D be a simple connected digraph with n vertices and m arcs and let W (D) = (D, w) be the weighted digraph corresponding to D, where the weights are taken from the set of non-zero real numbers. Let v1,V2,...,Vn be the eigenvalues of the skew Laplacian weighted matrix SLW (D) of the weighted digraph W(D). In this paper, we discuss the skew Laplacian energy SLEW(D) of weighted digraphs and obtain the skew Laplacian energy of the weighted star W(K1,n) for some fixed orientation to the weighted arcs. We obtain lower and upper bounds for SLEW(D) and show the existence of weighted digraphs attaining these bounds. [ABSTRACT FROM AUTHOR]

Let G be a connected graph and W = {w1;w2; . . . ;}kg be an ordered subset of vertices of G. For any vertex v of G, the ordered k-vector r(v|W) = (d(v;w1); d(v;w2); . . .; d(v;wk)) is called the metric representation of v with respect to W, where d(x; y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension denoted by dim(G). A resolving set W is called a non-isolated resolving set for G if the induced subgraph hWi of G has no isolated vertices. The minimum cardinality of a non-isolated resolving set for G is called the non-isolated resolving number of G and denoted by nr(G). The aim of this paper is to find properties of unicyclic graphs that have non-isolated resolving number 2 and then to characterize all these graphs. [ABSTRACT FROM AUTHOR]

This paper researches the fixed-time leader-following consensus problem for nonlinear multi-agent systems (MASs) affected by unknown disturbances under the jointly connected graph. In order to achieve control goal, this paper designs a fixed-time consensus protocol, which can offset the unknown disturbances and the nonlinear item under the jointly connected graph, simultaneously. In this paper, the states of multiple followers can converge to the state of the leader within a fixed time regardless of the initial conditions rather than just converging to a small neighborhood near the leader state. Finally, a simulation example is given to illustrate the theoretical result. [ABSTRACT FROM AUTHOR]

An st $st$‐ordering of a graph G=(V,E) $G=(V,E)$ is an ordering v1,v2,...,vn ${v}_{1},{v}_{2},\ldots ,{v}_{n}$ of its vertex set such that s=v1,t=vn $s={v}_{1},t={v}_{n}$ and every vertex vi ${v}_{i}$ with i=2,3,...,n−1 $i=2,3,\ldots ,n-1$ has both a lower numbered and a higher numbered neighbor. Such orderings have played an important role in algorithms for planarity testing. It is well‐known that every 2‐connected graph has an st $st$‐ordering for every choice of distinct vertices s,t $s,t$. An st $st$‐ordering of a graph G $G$ corresponds directly to a so‐called bipolar orientation of G $G$, that is, an acyclic orientation D $D$ of G $G$ in which s $s$ is the unique source and t $t$ is the unique sink. Clearly every bipolar orientation of a graph has an out‐branching rooted at the source vertex and an in‐branching rooted at the sink vertex. In this paper, we study graphs which admit a bipolar orientation that contains an out‐branching and in‐branching which are arc‐disjoint (such an orientation is called good). A 2T‐graph is a graph whose edge set can be decomposed into two edge‐disjoint spanning trees. Clearly a graph has a good orientation if and only if it contains a spanning 2T‐graph with a good orientation, implying that 2T‐graphs play a central role. It is a well‐known result due to Tutte and Nash‐Williams, respectively, that every 4‐edge‐connected graph contains a spanning 2T‐graph. Vertex‐minimal 2T‐graphs with at least two vertices, also known as generic circuits, play an important role in rigidity theory for graphs. Recently with Bessy and Huang we proved that every generic circuit has a good orientation. In fact, we may specify the roots of the two branchings arbitrarily as long as they are distinct. Using this, several results on good orientations of 2T‐graphs were obtained. It is an open problem whether there exists a polynomial algorithm for deciding whether a given 2T‐graph has a good orientation. Complex constructions of 2T‐graphs with no good orientation were given in work by Bang‐Jensen, Bessy, Huang and Kriesell (2021) indicating that the problem might be very difficult. In this paper, we focus on so‐called quartics which are 2T‐graphs where every vertex has degree 3 or 4. We identify a sufficient condition for a quartic to have a good orientation, give a polynomial algorithm to recognize quartics satisfying the condition and a polynomial algorithm to produce a good orientation when this condition is met. As a consequence of these results we prove that every 4‐regular and 4‐connected graph has a good orientation, where, as for generic circuits, we may specify the roots of the two branchings arbitrarily as long as they are distinct. We also provide evidence that even for quartics it may be difficult to find a characterization of those instances which have a good orientation. We also show that every graph on n≥8 $n\ge 8$ vertices and of minimum degree at least ⌊n∕2⌋ $\lfloor n\unicode{x02215}2\rfloor $ has a good orientation. Finally we pose a number of open problems. [ABSTRACT FROM AUTHOR]

In this paper we are interested in an intrinsic property of graphs which is derived from their embeddings into the Euclidean 3‐space R3. An embedding of a graph into R3 is said to be linear, if it sends every edge to be a line segment. And we say that an embedding f of a graph G into R3 is free, if π1(R3−f(G)) is a free group. Lastly a simple connected graph is said to be linearly free if every its linear embedding is free. In the 1980s it was proved that every complete graph is linearly free, by Nicholson. In this paper, we develop Nicholson's arguments into a general notion, and establish a sufficient condition for a linear embedding to be free. As an application of the condition we give a partial answer for a question: How much can the complete graph Kn be enlarged so that the linear‐freeness is preserved and the clique number does not increase? And an example supporting our answer is provided. As the second application it is shown that a simple connected graph of minimal valency at least 3 is linearly free, if it has less than eight vertices. The conditional inequality is strict, because we found a graph with eight vertices which are not linearly free. It is also proved that for n,m≤6 the complete bipartite graph Kn,m is linearly free. [ABSTRACT FROM AUTHOR]

In recent years, graph neural network algorithm (GNN) for graph semi-supervised classification has made great progress. However, in the task of node classification, the neighborhood size is often difficult to expand. The propagation of nodes always only considers the nearest neighbor nodes. Some algorithms usually approximately classify by message passing between direct (single-hop) neighbors. This paper proposes a simple and effective method, named Graph Mixed Random Network Based on PageRank (PMRGNN) to solve the above problems. In PMRGNN, we design a PageRank-based random propagation strategy for data augmentation. Then, two feature extractors are used in combination to supplement the mutual information between features. Finally, a graph regularization term is designed, which can find more useful information for classification results from neighbor nodes to improve the performance of the model. Experimental results on graph benchmark datasets show that the method of this paper outperforms several recently proposed GNN baselines on the semi-supervised node classification. In the research of over-smoothing and generalization, PMRGNN always maintains better performance. In classification visualization, it is more intuitive than other classification methods. [ABSTRACT FROM AUTHOR]

For any graph G of order n with degree sequence d 1 ≥ ⋯ ≥ d n , we define the double Slater number s ℓ × 2 (G) as the smallest integer t such that t + d 1 + ⋯ + d t - e ≥ 2 n - p in which e and p are the number of end-vertices and penultimate vertices of G, respectively. We show that γ × 2 (G) ≥ s ℓ × 2 (G) , where γ × 2 (G) is the well-known double domination number of a graph G with no isolated vertices. We prove that the problem of deciding whether the equality holds for a given graph is NP-complete even when restricted to 4-partite graphs. We also prove that the problem of computing γ × 2 (G) is NP-hard even for comparability graphs of diameter two. Some results concerning these two parameters are given in this paper improving and generalizing some earlier results on double domination in graphs. We give an upper bound on the k-tuple domatic number of graphs with characterization of all graphs attaining the bound. Finally, we characterize the family of all full graphs, leading to a solution to an open problem given in a paper by Cockayne et al. (Networks 7: 247–261, 1977). [ABSTRACT FROM AUTHOR]

In 1987, Alavi, Boals, Chartrand, Erdos, and Oellermann conjectured that all graphs have an ascending subgraph decomposition (ASD). In a previous paper, Wagner showed that all oriented complete balanced tripartite graphs have an ASD. In this paper, we will show that all orientations of an oriented graph that can be factored into triangles with a large portion of the triangles being transitive have an ASD. We will also use the result to obtain an ASD for any orientation of complete multipartite graphs with 3n partite classes each containing 2 vertices (a K(2: 3n)) or 4 vertices (a K(4: 3n)). [ABSTRACT FROM AUTHOR]

The objective of the present paper is to study the maximum radius r of a connected graph of order n, minimum degree δ≥2 and girth at least g≥4. Erdős, Pach, Pollack and Tuza proved that if g=4, that is, the graph is triangle‐free, then r≤n−2δ+12, and noted that up to the value of the additive constant, this upper bound is tight. In this paper we shall determine the exact maximum. For larger values of g little is known. We settle the order of the maximum r for g=6,8 and 12, and prove an upper bound for every even g, which we conjecture to be tight up to a constant factor. Finally, we show that our conjecture implies the so‐called Erdős girth conjecture. [ABSTRACT FROM AUTHOR]

Co-movement among individual firms' stock prices can reflect complex interfirm relationships. This paper proposes a novel method to leverage such relationships for stock price predictions by adopting inductive graph representation learning on dynamic stock graphs constructed based on historical stock price co-movement. To learn node representations from such dynamic graphs for better stock predictions, we propose the hybrid-attention dynamic graph neural network, an inductive graph representation learning method. We also extended mini-batch gradient descent to inductive representation learning on dynamic stock graphs so that the model can update parameters over mini-batch stock graphs with higher training efficiency. Extensive experiments on stocks from different markets and trading simulations demonstrate that the proposed method significantly improves stock predictions. The proposed method can have important implications for the management of financial portfolios and investment risk. Summary of Contribution: Accurate predictions of stock prices have important implications for financial decisions. In today's economy, individual firms are increasingly connected via different types of relationships. As a result, firms' stock prices often feature synchronous co-movement patterns. This paper represents the first effort to leverage such phenomena to construct dynamic stock graphs for stock predictions. We develop hybrid-attention dynamic graph neural network (HAD-GNN), an inductive graph representation learning framework for dynamic stock graphs to incorporate temporal and graph attention mechanisms. To improve the learning efficiency of HAD-GNN, we also extend the mini-batch gradient descent to inductive representation learning on such dynamic graphs and adopt a t-batch training mechanism (t-BTM). We demonstrate the effectiveness of our new approach via experiments based on real-world data and simulations. [ABSTRACT FROM AUTHOR]

Existing Web services recommendation models suffer from matrix sparsity and cold-start problems due to the limited number of user-service interactions and too little contextual information, so this paper introduces knowledge graph as auxiliary information to Web service recommendation to alleviate matrix sparsity and cold-start problems. And we propose the Web services recommendation based on Metapath-guided Graph Attention Network Model (WSR-MGAT) to fully exploit the structural information of the knowledge graph to improve the recommendation accuracy. Specifically, WSR-MGAT uses graph embedding method to obtain the initial embedding of entities and relationships. In order to obtain more closely related neighbors, we propose to use a distance-aware path sampling method to extract meta-path instances with closer relationships. Previous knowledge graph-based Web service recommendations do not make full use of the rich interaction information, which may lead to limited performance. To address this problem, this paper uses meta-paths to guide nodes to recursively aggregate higher-order neighbor information and use an attention mechanism to distinguish the importance of neighbors. Meanwhile, the semantic information of nodes under different meta-paths is fused to obtain a more comprehensive embedding of nodes. To verify the performance of the model, we use the real data crawled from the ProgrammableWeb platform to conduct multiple groups of experiments. The experimental results show that the WSR-MGAT improves over the strongest baselines w.r.t. Prec@10 by 4.6%; Recall@10 by 3.8%; NDCG@10 by 4.2% and F1@10 by 4.9%. [ABSTRACT FROM AUTHOR]

Lots of symmetric properties are well-explored and analyzed in extremal graph theory, such as the well-known symmetrization operation in the Turán problem and the high symmetric in the extremal graphs. This paper is devoted to studying the Lagrangian of hypergraphs, which connects to a very symmetric function—the Lagrangian function. Given an r-uniform hypergraph F, the Lagrangian density π λ (F) is the limit supremum of r ! λ (G) over all F-free G, where λ (G) is the Lagrangian of G. An r-uniform hypergraph F is called λ -perfect if π λ (F) equals r ! λ (K v (F) − 1 r) . Yan and Peng conjectured that: for integer r ≥ 3 , there exists n 0 (r) such that if G and H are two λ -perfect r-graphs with | V (G) | and | V (H) | no less than n 0 (r) , then the disjoint union of G and H is λ -perfect. Let S t denote a 3-uniform hypergraph with t edges { e 1 , ⋯ , e t } satisfying that e i ∩ e j = { v } for all 1 ≤ i < j ≤ t . In this paper, we show that the conjecture holds for G = S 2 and H = S t for all t ≥ 62 . Moreover, our result yields a class of Turán densities of 3-uniform hypergraphs. In our proof, we use some new techniques to study Lagrangian density problems; using a result of Sidorenko to find subgraphs, and a result of Talbot to upper bound the Lagrangian of a hypergraph. [ABSTRACT FROM AUTHOR]