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The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted I (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. We show that not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a seed graph G such that H ≅ I (G). Examples of such graphs include K4 − e and K2,3. We build a series of tools to show that known i-graphs can be used to construct new i-graphs and apply these results to build other classes of i-graphs, such as block graphs, hypercubes, forests, cacti, and unicyclic graphs. [ABSTRACT FROM AUTHOR]

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]

We prove a version of the strong Taylor's conjecture for stable graphs: if G is a stable graph whose chromatic number is strictly greater than Æ2.@0/then G contains all finite subgraphs of Shn.!/and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model-theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability. [ABSTRACT FROM AUTHOR]

We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high enough cardinality will have a single model in any high enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation. [ABSTRACT FROM AUTHOR]

We consider the problem of counting the number of copies of a fixed graph H within an input graph G. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input G has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g., planar graphs), as well as graphs generated by various random processes (e.g., preferential attachment graphs). We say that H is easy if there is a linear-time algorithm for counting the number of copies of H in an input G of bounded degeneracy. A seminal result of Chiba and Nishizeki from '85 states that every H on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all H on 5 vertices and further proved that for every k > 5 there is a k-vertex H which is not easy. They left open the natural problem of characterizing all easy graphs H. Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph H to be easy. Here, we show that this sufficient condition is also necessary, thus fully answering the Bera-Pashanasangi-Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms. [ABSTRACT FROM AUTHOR]

We consider the evaluation of first-order queries over classes of databases that are nowhere dense. The notion of nowhere dense classes was introduced by Nešetřil and Ossona de Mendez as a formalization of classes of "sparse" graphs and generalizes many well-known classes of graphs, such as classes of bounded degree, bounded tree-width, or bounded expansion. It has recently been shown by Grohe, Kreutzer, and Siebertz that over nowhere dense classes of databases, first-order sentences can be evaluated in pseudo-linear time (pseudo-linear time means that for all - there exists an algorithm working in time O(n1+ε), where n is the size of the database). For first-order queries of higher arities, we show that over any nowhere dense class of databases, the set of their solutions can be enumerated with constant delay after a pseudo-linear time preprocessing. In the same context, we also show that after a pseudo-linear time preprocessing we can, on input of a tuple, test in constant time whether it is a solution to the query. [ABSTRACT FROM AUTHOR]

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA'14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, Kt -free unit d-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d-contractions, witness that the twin-width is at most d. We show that FO model checking, that is deciding if a given first-order formula ϕ evaluates to true for a given binary structure G on a domain D, is FPT in |ϕ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d-contraction sequence for G, our algorithm runs in time f (d, |ϕ|) · |D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved under FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS'15]. [ABSTRACT FROM AUTHOR]

In the k-cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O(n2k) time. However, lower bounds from conjectures about the k-clique problem imply that O(n(1-o(1))k) time is likely needed. Recent results of Gupta, Lee, and Li have given new algorithms for general k-cut in n1.98k+O(1) time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In thiswork, we resolve the problem for general graphs. We showthat the Contraction Algorithm of Karger outputs any fixed k-cut of weight a-k with probability Ok (n -ak), where -k denotes the minimum k-cut weight. This also gives an extremal bound of Ok (nk) on the number of minimum k-cuts and an algorithm to compute -k with roughly nk polylog(n) runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k-clique. The firstmain ingredient in our result is an extremal bound on the number of cuts of weight less than 2-k/k, using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks--and how the average degree evolves--in the Karger process. [ABSTRACT FROM AUTHOR]

We describe various diagram algebras and their representation theory using cellular algebras of Graham and Lehrer and the decomposition into half diagrams. In particular, we show the diagram algebras surveyed here are all cellular algebras and parameterize their cell modules. We give a new construction to build new cellular algebras from a general cellular algebra and subalgebras of the rook Brauer algebra that we call the cellular wreath product. [ABSTRACT FROM AUTHOR]

Much research has reported on students' difficulties with solving related rates problems in calculus. In an effort to generate a resource that could potentially address some of these difficulties from a teaching standpoint, a questionnaire about effective instructional approaches related to the teaching of related rates problems, among other things, was administered to 14 veteran calculus instructors. Analysis of the responses provided by the instructors revealed that all the instructors considered the use of diagrams to be helpful when solving related rates problems. Furthermore, a majority of these instructors noted that introducing a set of steps (i.e., a guideline), during classroom instruction, that students could follow when solving related rates problems is helpful for students when working with this type of problems. These instructors further identified strengths and weaknesses in the way related rates problems are typically presented in calculus textbooks. Implications for instruction are included. [ABSTRACT FROM AUTHOR]

The natural language to SQL (NL2SQL) task is an emerging research area that aims to transform a natural language with a given database into an SQL query. The earlier approaches were to process the input into a heterogeneous graph. However, previous models failed to distinguish the types of multi-hop connections of the heterogeneous graph, which tended to ignore crucial semantic path information. To this end, a two-layer attention network is presented to focus on essential neighbor nodes and mine enlightening semantic paths for feature encoding. The weighted edge is introduced for schema linking to connect the nodes with semantic similarity. In the decoding phase, a rule-based pruning strategy is offered to refine the generated SQL queries. From the experimental results, the approach is shown to learn a good encoding representation and decode the representation to generate results with practical meaning. [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]

The generalized dynamic Tullock contest model with two homogeneous participants is established, in which both players have the same valuation of winning rewards and losing rewards. Firstly, the unique symmetric equilibrium point of the system is obtained by calculation and its local stability condition is given based on the Jury criterion. Then, two paths of the system from stability to chaos, namely flip bifurcation and Neimark-Sacker bifurcation, are analyzed by using the two-dimensional parametric bifurcation diagram. Meanwhile, the abundant Arnold tongues in the two-dimensional parametric bifurcation diagram are analyzed. Finally, the phenomenon of multistability of the system is illustrated through the basin of attraction, and the contact bifurcation occurs during the evolution of the basin of attraction with varying parameters. [ABSTRACT FROM AUTHOR]

Based on DBR, a pressure vessel and standard vessel are designed using a public view and diagrams. They are formulated to a standard specification using the ASME pressure vessel code rules and the European standard EN13445. The pressure is determined under influence of a closed end. There are limitations to the structure of the container. Using DBA, several analyses were performed based on FEM to provide parameters to complete the DOE, and combined with DFA and DFM to verify that the analysis was conducive to giving users a good analysis plan. The core premise of this study is to simplify operation, to provide an exemplary user interface, observe various parameters according to requirements, meet pressure requirements during the reaction of polymer composite materials, and ensure that the temperature, strength, and manufacturing cost are within safety factors, to provide a good user experience. [ABSTRACT FROM AUTHOR]

A simple theoretical model of network evolution is discussed here. In each step, we add a new vertex to the graph and it is allowed to connect it to maximum degree vertices (hubs) only. Given a constant p, the probability of such a connection is p for any hub. The initial (non-random) graph G1 is arbitrary but here we investigate mostly the case when G1 has one vertex. We solve here some particular cases of the problem, using enumeration methods. We obtain not limit theorems but exact results for the parameters discussed. [ABSTRACT FROM AUTHOR]

Cop Robber game is a two player game played on an undirected graph. In this game, the cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. We study cop-edge critical graphs, i.e. graphs G such that for any edge e in E(G) either c(G -e) < c(G) or c(G-e) > c(G). In this article, we study the edge criticality of generalized Petersen graphs and Paley graphs. [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 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 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]

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]

The 2S 3 transformation, which was first described for positive integers, has been defined for dyadic rational numbers in the op en interval (0, 1) in this study. The set of dyadic rational numbers is a Prüfer 2-group. For the dyadic 2S3 transformation Tds(x), the restricted multiplicative and additive properties have been established. Graph parameters are used to generate more combinatorial outcomes for these properties. The relationship between the SM dyadic sum graph's automorphism group and the symmetric group has been investigated. [ABSTRACT FROM AUTHOR]

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]

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]

The Sombor index of the graph G is a degree based topological index, defined as SO=∑uv∈E(G)d²u+d²v--√, where du is the degree of the vertex u, and E(G) is the edge set of G. Bounds on SO are established in terms of graph energy, size of minimum vertex cover, matching number, and induced matching number. [ABSTRACT FROM AUTHOR]

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We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-4 internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations, we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space M0;4. We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations. [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]

The path energy is a recently conceived variant of graph energy, equal to the sum of the absolute values of the eigenvalues of the path matrix [Shikare et. al, 2018]. Two graphs having equal path energy are said to be path-equienergetic. All trees of same order are mutually path-equienergetic. Members of a large class of connected regular graphs of same order and degree are mutually path-equienergetic. In this paper, we construct other types of path-equienergetic graphs and explore their spectral properties. [ABSTRACT FROM AUTHOR]

We say that a regular graph G of order n and degree r ⩾ 1 (which is not the complete graph) is strongly regular if there exist non-negative integers τ and θ such that |Si ∩ Sj | = τ for any two adjacent vertices i and j, and |Si ∩ Sj | = θ for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let λ1 = r, λ2 and λ3 be the distinct eigenvalues of a connected strongly regular graph. Let m1 = 1, m² and m³ denote the multiplicity of r, λ2 and λ3, respectively. We here describe the parameters n, r, τ and θ for strongly regular graphs with m2 = qm3 and m3 = qm2 for q = 9, 10. [ABSTRACT FROM AUTHOR]

Gutman introduced the so-called Sombor index, a unique degreebased graphical index having geometrical importance. In a similar manner, Kulli extended the neighborhood version of the Sombor index for the graph G. In this paper, several bounds and characterizations are obtained. Along with the chemical application in nitro polycyclic aromatic hydrocarbons (nitro-PAHs), several relationships between the neighbor Sombor index and other degree and neighbor sum based graphical indices are also obtained. [ABSTRACT FROM AUTHOR]

Existing approaches to indoor environmental quality in single-family residential buildings have been described, identifying their optimal preferred parameters for maintaining a human-healthy environment. The energy requirements for maintaining basic building systems (lighting, cooking, heating, ventilation, cooling) have been analyzed. Passive and active sources of energy production have been analyzed, prosumer buildings have been considered, and methods of obtaining maximum energy in winter periods and avoiding overheating of the building in summer periods have been discussed. Process schemes of using active shadow structures as an integral element with other systems acquiring energy from RES in single-family residential buildings have been presented. Architectural aspects of the use of energy-active shadow structures in residential buildings (single – family houses) have been presented together with alternative solutions. This paper presents process diagrams of the use of energy-active shadow integrated with other renewable energy sources used in single-family buildings, such as: BIPV, PVT and heat pumps, benefiting indoor environmental quality. Architectural considerations for the use of energy-active shadow structures in single-family homes have been presented along with alternative solutions. The purpose of this paper is to present a concept (especially effective in the summer) for an additional way to harvest electricity from energy-active shadows in single-family houses. [ABSTRACT FROM AUTHOR]

For a graph G, a spanning subgraph F of G is called a {P2,P5}-factor if every component of F is isomorphic to P2 or P5, where Pi denotes the path of order i. A graph G is called a ({P2,P5},k)-factor critical graph if G-V' contains a {P2,P5}-factor for any V' ⊆V(G) with |V'| = k. A graph G is called a ({P2,P5},m)- factor deleted graph if G-E' has a {P2,P5}-factor for any E' ⊆ E(G) with |E'| = m. The degree sum of G is defined by σr+1(G) = ... min X⊆V(G) n S x'X dG(x): X is an independent set of r+1 vertices }. In this paper, using degree sum conditions, we demonstrate that (i) G is a ({P2,P5},k)-factor critical graph if σr+1(G) > >(3n+4k-2)(r+1) 7 and ≥(G) = k+r; (ii) G is a ({P2,P5},m)-factor deleted graph if σ2+1(G) > (3n+2m-2)(r+1) 7 and ≥(G) = 5m 4 +r. [ABSTRACT FROM AUTHOR]

The Wiener index and the Mostar index quantify two distance related properties of connected graphs: the Wiener index is the sum of the distances over all pairs of vertices and the Mostar index is a measure of how far the graph is from being distance-balanced. These two measures have been considered for a number of interesting families of graphs. In this paper, we determine the Wiener index and the Mostar index of alternate Lucas cubes. Alternate Lucas cubes form a family of interconnection networks whose recursive construction mimics the construction of the well-known Fibonacci cubes. [ABSTRACT FROM AUTHOR]

The Sombor index of the graph G is a degree based topological index, deffned as SO =P uv2E(G) p dG(u)2 + dG(v)2, where dG(u) is the degree of the vertex u of G and E(G) is the edge set of G. In this paper we calculate SO of some graph transformations. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with n vertices. The Kirchhoff index of G is defined as Kf (G) = n Pn1 i=1 > n = 0 are the Laplacian eigenvalues of G. Some bounds on Kf (G) in terms of graph parameters such as the number of vertices, the number of edges, first Zagreb index, forgotten topological index, etc., are presented. These bounds improve some previously known bounds in the literature. [ABSTRACT FROM AUTHOR]

The index ≥1(1) of a signed graph 1= (G; -) is just the largest eigenvalue of its adjacency matrix. For any n > 4 we identify the signed graphs achieving the minimum index in the class of signed bicyclic graphs with n vertices. Apart from the n = 4 case, such graphs are obtained by considering a starlike tree with four branches of suitable length (i.e. four distinct paths joined at their end vertex u) with two additional negative independent edges pairwise joining the four vertices adjacent to u. As a by-product, all signed bicyclic graphs containing a theta-graph and whose index is less than 2 are detected. [ABSTRACT FROM AUTHOR]

A 2-rainbow dominating function on a graph G is a function g that assigns to each vertex a set of colors chosen from the subsets of f1; 2g so that for each vertex with g(v) =; we have S u2N(v) g(u) = f1; 2g. The weight of a 2-rainbow dominating function g is the value w(g) = P v2V (G) jf(v)j. A 2-rainbow dominating function g is an independent 2-rainbow dominating function if no pair of vertices assigned nonempty sets are adjacent. The 2-rainbow domination number r2(G) (respectively, the inde-pendent 2-rainbow domination number ir2(G)) is the minimum weight of a 2-rainbow dominating function (respectively, independent 2-rainbow dominating function) on G. We prove that for any tree T of order ≥ 3, with l leaves and s support vertices, ir2(T) ≤ (14n + ' + s)=20, thus improving the bound given in [Independent 2-rainbow domination in trees, Asian-Eur. J. Math. 8 (2015) 1550035] under certain conditions. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G). An Italian dominating function (IDF) is a function f: V (G) ! f0; 1; 2g having the property that that f(N(u)) ≥ 2 for every vertex u 2 V (G) with f(u) = 0, where N(u) is the neighborhood of u. If f is an IDF on G, then let V0 = fv 2 V (G): f(v) = 0g. A restrained Italian dominating function (RIDF) is an Italian dominating function f having the property that the subgraph induced by V0 does not have an isolated vertex. The weight of an RIDF f is the sum P v2V (G) f(v), and the minimum weight of an RIDF on a graph G is the restrained Italian domination number. We present sharp bounds for the restrained Italian domination number, and we determine the restrained Italian domination number for some families of graphs. [ABSTRACT FROM AUTHOR]

Let G = (V;E) be a given graph of order n. A function f: V ! f0; 1; 2g is an independent Roman dominating function (IRDF) on G if for every vertex v 2 V with f(v) = 0 there is a vertex u adjacent to v with f(u) = 2 and fv 2 V: f(v) > 0g is an independent set. The weight of an IRDF f on G is the value f(V) = P v2V f(v). The minimum weight of an IRDF among all IRDFs on G is called the independent Roman domination number of G. In this paper, we give algorithms for computing the independent Roman domination number of G in O(jV j) time when G = (V;E) is a tree, unicyclic graph or proper interval graph. [ABSTRACT FROM AUTHOR]

The annihilating-ideal graph of a commutative ring R with unity is deffned as the graph AG(R) whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices I and J are adjacent if and only if IJ = 0. Nikan-dish et.al. proved that AG(Zn) is weakly perfect. In this short paper, we characterize n for which AG(Zn) is perfect. [ABSTRACT FROM AUTHOR]

Stress is an important centrality measure of graphs applicable to the study of social and biological networks. We study the stress of paths, cycles, fans and wheels. We determine the stress of a cut vertex of a graph G; when G has at most two cut vertices. We have also identified the graphs with minimum stress and maximum stress in the family of all trees of order n and in the family of all complete bipartite graphs of order n. [ABSTRACT FROM AUTHOR]

For a finite commutative ring Zn with identity 1 6= 0, the zero divisor graph (Zn) is a simple connected graph having vertex set as the set of non-zero zero divisors, where two vertices x and y are adjacent if and only if xy = 0. We find the Randic spectrum of the zero divisor graphs (Zn), for various values of n and characterize n for which (Zn) is Randić integral. [ABSTRACT FROM AUTHOR]

Let G be a fuzzy graph of order n, where(u) is the vertex membership, (u; v) is membership value of an edge and (u) is the strength of vertex. The first fuzzy Zagreb index is the sum -(ui)-(ui) + -(uj)-(uj) where uiuj 2 - and the corresponding fuzzy Zagreb matrix is the square matrix of order n whose (i; j)th entry whenever i 6= j, is -(ui)-(ui)+-(uj)-(uj) and zero otherwise. In this paper, we introduce the Zagreb Estrada index of fuzzy graphs and establish some bounds for it. [ABSTRACT FROM AUTHOR]

The Randić index R(G) of a graph G is the sum of the weights (dudv)1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randić indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex unicyclic graphs with the third maximum for n ≥ 5, the fourth maximum for n ≥ 7 and the fifth maximum for n ≥ 8. Recently, Li et al. [The Randić indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex unicyclic graphs with the sixth maximum and the seventh maximum for n ≥ 9 and the eighth maximum for n ≥ 10. In this paper, we characterize the n-vertex unicyclic graphs with the ninth maximum, the tenth maximum, the eleventh maximum, the twelfth maximum and the thirteenth maximum of Randić values. [ABSTRACT FROM AUTHOR]