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Background:: A topological index of a molecular graph is the numeric quantity which can predict certain physical and chemical properties of the corresponding molecule. Xu et al. introduced some graph transformations which increase or decrease the first and second multiplicative Zagreb indices and proposed a unified approach to characterize extremal (n, m)- graphs. Method:: Graph transformations are used to find the extremal graphs, these transformations either increase or decrease the general multiplicative Zagreb indices. By applying the transformations which increase the general multiplicative Zagreb indices we find the graphs with maximal general multiplicative Zagreb indices and for minimal general Zagreb indices we use the transformations which decrease the index. Result:: In this paper, we extend the Xu’s results and show that the same graph transformations increase or decrease the first and second general multiplicative Zagreb indices for . As an application, the extremal acyclic, unicyclic and bicyclic graphs are presented for general multiplicative Zagreb indices. Conclusion:: By applying the transformation we investigated that in the class of acyclic, unicyclic and bicyclic graphs, which graph gives the minimum and the maximum general multiplicative Zagreb indices.

The Kirchhoff index and degree-Kirchhoff index have attracted extensive attentions due to their wide applications in physics and chemistry. These indices have been computed for many interesting graphs, such as linear polyomino chain, linear/Mobius/cylinder hexagonal chain, and linear octagonal chain. In the present paper, we consider Mobius octagonal chain ( M n ) and cylinder octagonal chain ( M n ′ ). Explicit closed-form formulae of the Kirchhoff index and the number of spanning trees are obtained for M n and M n ′ .

For the study and valuation of social graphs, which affect an extensive range of applications such as community decision-making support and recommender systems, it is highly recommended to sustain the resistance of a social graph G to active attacks. In this regard, a novel privacy measure, called the k , l -anonymity, is used since the last few years on the base of k -metric antidimension of G in which l is the maximum number of attacker nodes defining the k -metric antidimension of G for the smallest positive integer k . The k -metric antidimension of G is the smallest number of attacker nodes less than or equal to l such that other k nodes in G cannot be uniquely identified by the attacker nodes. In this paper, we consider four families of wheel-related social graphs, namely, Jahangir graphs, helm graphs, flower graphs, and sunflower graphs. By determining their k -metric antidimension, we prove that each social graph of these families is the maximum degree metric antidimensional, where the degree of a vertex is the number of vertices linked with that vertex.

In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by Y α G = ∑ u v ∈ E G d u d u + d v d v α , where d u and d v represent the degree of vertices u and v , respectively, and α ≥ 1 . A connected graph G is called a k -generalized quasi-tree if there exists a subset V k ⊂ V G of cardinality k such that the graph G − V k is a tree but for any subset V k − 1 ⊂ V G of cardinality k − 1 , the graph G − V k − 1 is not a tree. In this work, we find a sharp lower and some sharp upper bounds for this new sum-connectivity index.

Let G = G 1 × G 2 × ⋯ × G m be the strong product of simple, finite connected graphs, and let ϕ : ℕ ⟶ 0 , ∞ be an increasing function. We consider the action of generalized maximal operator M G ϕ on ℓ p spaces. We determine the exact value of ℓ p -quasi-norm of M G ϕ for the case when G is strong product of complete graphs, where 0 < p ≤ 1 . However, lower and upper bounds of ℓ p -norm have been determined when 1 < p < ∞ . Finally, we computed the lower and upper bounds of M G ϕ p when G is strong product of arbitrary graphs, where 0 < p ≤ 1 .

Sierpiński graphs are family of fractal nature graphs having applications in mathematics of Tower of Hanoi, topology, computer science, and many more diverse areas of science and technology. This family of graphs can be generated by taking certain number of copies of the same basic graph. A topological index is the number which shows some basic properties of the chemical structures. This article deals with degree based topological indices of uniform subdivision of the generalized Sierpiński graphs S(n,G) and Sierpiński gasket Sn . The closed formulae for the computation of different kinds of Zagreb indices, multiple Zagreb indices, reduced Zagreb indices, augmented Zagreb indices, Narumi-Katayama index, forgotten index, and Zagreb polynomials have been presented for the family of graphs.

The 2-metric resolvability is an extension of metric resolvability in graphs having several applications in intelligent systems for example network optimization, robot navigation and sensor networking. Rotationally symmetric graphs are important in intelligent networks due to uniform rate of data transformation to all nodes. In this article, 2-metric dimension of rotationally symmetric plane graphs Rn, Sn and Tn is computed and found to be independent of the number of vertices.

Two graphs are said to be cospectral with respect to the Laplacian matrix if they have the same Laplacian spectrum. A graph is said to be determined by the Laplacian spectrum if there is no other non-isomorphic graph with the same Laplacian spectrum. In this paper, we prove that one special class of triple starlike tree is determined by its Laplacian spectrum.

In this work, we incorporate modular arithmetic and discuss a special class of graphs based on power functions in a given modulus, called power digraphs. In power digraphs, the study of cyclic structures and enumeration of components is a difficult task. In this manuscript, we attempt to solve the problem for $ p $th power congruences over different classes of residues, where $ p $ is an odd prime. For any positive integer $ m $, we build a digraph $ G(p, m) $ whose vertex set is $ \mathbb{Z}_{m} = \{0, 1, 2, 3, ..., m-1\} $ and there will be a directed edge from vertices $ u\in \mathbb{Z}_{m} $ to $ v\in \mathbb{Z}_{m} $ if and only if $ u^{p}\equiv v\; (\text{mod} \; m) $. We study the structures of $ G(p, m) $. For the classes of numbers $ 2^{r} $ and $ p^{r} $ where $ r\in \mathbb{Z^{+}} $, we classify cyclic vertices and enumerate components of $ G(p, m) $. Additionally, we investigate two induced subdigraphs of $ G(p, m) $ whose vertices are coprime to $ m $ and not coprime to $ m $, respectively. Finally, we characterize regularity and semiregularity of $ G(p, m) $ and establish some necessary conditions for cyclicity of $ G(p, m) $.

For a graph G , an ordered set S ⊆ V G is called the resolving set of G , if the vector of distances to the vertices in S is distinct for every v ∈ V G . The minimum cardinality of S is termed as the metric dimension of G . S is called a fault-tolerant resolving set (FTRS) for G , if S \ v is still the resolving set ∀ v ∈ V G . The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of G . Due to enormous application in science such as mathematics and computer, the notion of the resolving set is being widely studied. In the present article, we focus on determining the FTMD of a generalized wheel graph. Moreover, a formula is developed for FTMD of a wheel and generalized wheels. Recently, some bounds of the FTMD of some of the convex polytopes have been computed, but here we come up with the exact values of the FTMD of two families of convex polytopes denoted as D k for k ≥ 4 and Q k for k ≥ 6 . We prove that these families of convex polytopes have constant FTMD. This brings us to pose a natural open problem about the existence of a polytope having nonconstant FTMD.

Graph theory plays a vital role in modeling and designing any chemical structure or chemical network. Chemical graph theory helps to understand the molecular structural properties of a molecular graph. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this paper, we have computed the various types of degree-based fifth M-Zagreb indices, general fifth M-Zagreb indices, fifth hyper-M-Zagreb indices, general fifth M1-Zagreb polynomial, general fifth M2-Zagreb polynomial, fifth M1-Zagreb polynomial and fifth M2-Zagreb polynomial, fifth hyper-M1-Zagreb polynomial, fifth hyper-M2-Zagreb polynomial and fifth M3-Zagreb index and their polynomials of a molecular graph namely carbon graphite denoted by CG(m,n) for t-levels.

A vertex $u\in V(G)$ resolves (distinguish or recognize) two elements (vertices or edges) $v,w\in E(G) \cup V(G)$ if $d_{G}(u,v)\neq d_{G}(u,w)$ . A subset $L_ {\textrm {m}}$ of vertices in a connected graph $G$ is called a mixed metric generator for $G$ if every two distinct elements (vertices and edges) of $G$ are resolved by some vertex set of $L_ {\textrm {m}}$ . The minimum cardinality of a mixed metric generator for $G$ is called the mixed metric dimension and is denoted by $dim_{m}(G)$ . In this paper, we studied the mixed metric dimension for three families of graphs $\mathcal {D}_{n}$ , $\mathcal {A}_{n}$ , and $\mathcal {R}_{n}$ , known from the literature. We proved that, for $\mathcal {D}_{n}$ the $dim_{m}(\mathcal {D}_{n})=dim_{e}(\mathcal {D}_{n})=dim(\mathcal {D}_{n})$ , when $n$ is even, and for $\mathcal {A}_{n}$ the $dim_{m}(\mathcal {A}_{n})=dim_{e}(\mathcal {A}_{n})$ , when $n$ is even and odd. The graph $\mathcal {R}_{n}$ has mixed metric dimension 5.

In this paper, we propose a family of nested weighted [Formula: see text]-polygon networks, which is a kind of promotion of infinite fractal dimension networks. We study the coherence of the networks with recursive features that contain the initial states dominated by a weighted parameter. The network coherence is a consensus problem with additive noises, and it is known that coherence is defined by the eigenvalues of the Laplacian matrix. According to the structure of recursive nested model, we get the recursive expressions of Laplacian eigenvalues and further derive the exact results of first- and second-order coherence. Finally, we investigate the influential impacts on the coherence for different large parameters and discuss the relationship between Laplacian energy and network coherence. Furthermore, we obtain the expressions of Kirchhoff index, mean first-passage time and average path length of the networks.

Group theory, the ultimate theory for symmetry, is a powerful tool that has a direct impact on research in robotics, computer vision, computer graphics and medical image analysis. Symmetry is very important in chemistry research and group theory is the tool that is used to determine symmetry. Usually, it is not only the symmetry of molecule but also the symmetries of some local atoms, molecular orbitals, rotations and vibrations of bonds, etc. that are important. Harada-Norton group is an example of a sporadic simple group. There are 14 maximal subgroups of Harada-Norton group. Generators (also known as words) of 11 maximal subgroups are already known. The aim of this note is to give generators of the remaining 3 maximal subgroups, which is an open problem mentioned on A World-wide-web Atlas of Group Representations (http://brauer.maths.qmul.ac.uk/Atlas) [1]. In this report we compute the generators of A6 × A6.D8, 23+2+6.(3 × L3(2)) and 34 : 2.(A4 × A4).4. Moreover we also compute the generators for the Maximal subgroups of some linear groups.

In this paper, we study resolvability and fault-tolerant resolvability of convex polytopes and related geometric graphs. Imran et al. (2010) [18] raised an open problem asserting that whether or not every family of convex polytope has a constant metric dimension. Raza et al. (2018) [28] studied the fault-tolerant metric dimension of certain families of convex polytopes. They also concluded their study with an open problem asking whether or not every family of convex polytope has a constant fault-tolerant metric dimension. In this paper, we provide negative answers to both of the aforementioned open problems. We answer the first question by constructing a family of convex polytopes with an unbounded metric dimension. By proving a result between resolvability and fault-tolerant resolvability structures of a graph, we show that the family of convex polytope with an unbounded metric dimension also possesses an unbounded fault-tolerant resolvability structure. Moreover, we construct three more infinite families of graphs which are closely related to convex polytopes, having an unbounded metric dimension.

Resistance distance is a novel distance function, also a new intrinsic graph metric, which makes some extensions of ordinary distance. Let $$O_n$$ be a linear crossed octagonal graph. Recently, Pan and Li (Int J Quantum Chem 118(24):e25787, 2018) derived the closed formulas for the Kirchhoff index, multiplicative degree-Kirchhoff index and the number of spanning trees of $$H_n$$. They pointed that it is interesting to give the explicit formulas for the Kirchhoff and multiplicative degree-Kirchhoff indices of $$O_n$$. Inspired by these, in this paper, two resistance distance-based graph invariants, namely, Kirchhoff and multiplicative degree-Kirchhoff indices are studied. We firstly determine formulas for the Laplacian (normalized Laplacian, resp.) spectrum of $$O_n$$. Further, the formulas for those two resistance distance-based graph invariants and spanning trees are given. More surprising, we find that the Kirchhoff (multiplicative degree-Kirchhoff, resp.) index is almost one quarter to Wiener (Gutman, resp.) index of a linear crossed octagonal graph.

The Sanskruti index of a graphGis defined as$$\begin{align*}S(G)=\sum_{uv\in{}E(G)}{\left(\frac{s_G(u)s_G(v)}{s_G(u)+s_G(v)-2}\right)}^3, \end{align*}$$wheresG(u) is the sum of the degrees of the neighbors of a vertexuinG. LetPn,Cn,SnandSn+ebe the path, cycle, star and star plus an edge ofnvertices, respectively. The Sanskruti index of a molecular graph of a compounds can model the bioactivity of compounds, has a strong correlation with entropy of octane isomers and its prediction power is higher than many existing topological descriptors.In this paper, we investigate the extremal trees and unicyclic graphs with respect to Sanskruti index. More precisely, we show that(1)$\frac{512}{27}n-\frac{172688}{3375}\leq{}S(T)\leq{}\frac{(n-1)^7}{8(n-2)^3}$for ann-vertex treeTwithn≤ 3, with equalities if and only ifT ≌Pn(left) andT≌Sn(right);(2)$ \frac{512}{27}n\leq{}S(G)\leq{}\frac{(n-3)(n+1)^3}{8}+\frac{3(n+1)^6}{8n^3}$for ann-vertex unicyclic graph withn≥ 4, with equalities if and only ifG ≌Cn(left) andG≌Sn+e(right).

The partition dimension problem is to decompose the vertex set of a graph into minimum number of vertex disjoint sets such that the ordered distances between every vertex of the graph and the disjoint sets of that decomposition have different values in the representation. This problem has emerging applications in areas such as structure-activity issues in drug design, navigation of robots, pattern recognition and image processing. In the present study, we provide an upper bound for the partition dimension of parallel composition of any graphs and derive an exact result for parallel composition of paths with different lengths. On the other side, we find the partition dimension of the series composition of cycles and complete graphs with different sizes and an upper bound is derived for series composition of any graphs. Further, this problem is applied to a graph which has been generated by parallel and series composition together on paths.

Let γ r 2 ( G ) be the 2-rainbow domination number of a graph G . In our work, we solve an open question for 2-rainbow domination number of general Petersen graphs P ( n , k ) . In addition, we proved that γ r 2 ( P ( n , k ) ) = n for n ≤ 12 , γ r 2 ( P ( n , 1 ) ) = n for n ≥ 5 , and γ r 2 ( P ( 2 k + 2 , k ) ) = 2 k + 2 for k ≥ 2 .

Recently, Peng and Li (2017) derived an explicit closed formula of Kirchhoff index and the number of spanning trees of linear phenylenes and their dicyclobutadieno derivatives, respectively in terms of the Laplacian spectrum. At the same time, they pointed that it is natural and interesting to study the normalized Laplacian, degree-Kirchhoff index and counting the spanning tree numbers of linear phenylenes and their dicyclobutadieno derivatives, respectively. Motivated by these, in this paper, explicit closed-form formulas for degree-Kirchhoff index and the number of spanning trees of generalized phenylenes are obtained based on the normalized Laplacian spectrum, respectively.

Mathematical modeling or numerical coding of the molecular structures play a significant role in the studies of the quantitative structure-activity relationships (QSAR) and quantitative structures property relationships (QSPR). In 1972, the entire energy of π-electron of a molecular graph is computed by the addition of square of degrees (valencies) of its vertices (nodes). Later on, this computational result was called by the first Zagreb index and became well studied topological index in the field of molecular graph theory. In this paper, for k ∈ N (set of counting numbers), we define four subdivision-related operations of graphs in their generalized form named by Sk, Rk, Qk and Tk. Moreover, using these operations and the concept of the cartesian product of graphs, we construct the generalized Fk-sum graphs Γ1+Fk Γ2, where Fk ∈ {Sk, Rk, Qk, Tk} and Γi are any connected graphs for i ∈ {1, 2}. Finally, the first and second Zagreb indices are computed for the generalized Fk-sum graphs in terms of their factor graphs. In fact, the obtained results are a general extension of the results Eliasi et al. and Deng et al. who studied these operations for exactly k = 1 and computed the Zagreb indices for only F1-sum graphs respectively.

The position of a moving point in a connected graph can be identified by computing the distance from the point to a set of sonar stations which have been appropriately situated in the graph. Let Q = {q1, q2, ... , qk} be an ordered set of vertices of a graph G and a is any vertex in G, then the code/representation of a w.r.t Q is the k-tuple (r(a, q1), r(a, q2), ... , r(a, qk)), denoted by r(a|Q). If the different vertices of G have the different representations w.r.t Q, then Q is known as a resolving set/locating set. A resolving/locating set having the least number of vertices is the basis for G and the number of vertices in the basis is called metric dimension of G and it is represented as dim(G). In this paper, the metric dimension of Toeplitz graphs generated by two and three parameters denoted by Tn〈1, t〉 and Tn〈1, 2, t〉, respectively is discussed and proved that it is constant.

Let Q(G) be the graph derived from G by inserting a new vertex into every edge of G and by connecting edges of these new vertices that are on adjacent edges of G, and Q-double join of G, G1 and G2 denoted by GQ v{G1, G2}, is the graph derived from Q(G), G1 and G2, by connecting every old-vertex V (G) of Q(G) with every vertex of G1 and every new-vertex I(G) of Q(G) with every vertex of G2. In this paper, we mainly considered the resistance distance and Kirchhoff index of Q-double join graphs GQ v {G1, G2} of an r-regular graph G and any two graphs of G1 and G2.

The numerical coding of the molecular structures on the bases of topological indices plays an important role in the subject of Cheminformatics which is a combination of Computer, Chemistry, and Information Science. In 1972, it was shown that the total π-electron energy of a molecular graph depends upon its structure and it can be obtained by the sum of the square of degrees of the vertices of a molecular graph. Later on, this sum was named as the first Zagreb index. In 2005, for γεR - {0, 1}, the first general Zagreb index of a graph G is defined as Mγ(G) = ΣvεV(G)[dG(v)]γ, where dG(v) is degree of the vertex v in G. In this paper, for each γεR - {0, 1}, we study the first general Zagreb index of the cartesian product of two graphs such that one of the graphs is D-sum graph and the other is any connected graph, where D-sum graph is obtained by using certain D operations on a connected graph. The obtained results are general extensions of the results of Deng et al. [Applied Mathematics and Computation 275(2016): 422-431] and Akhter et al. [AKCE Int. J. Graphs Combin. 14(2017): 70-79] who proved only for γ = 2 and γ = 3, respectively.

Graph labeling has wide applications in the field of computer science, such as coding theory, cryptography, software testing, database management systems, computer architecture, and networking. The computers connected in a network can now be converted in a graph and labels assigned to the graph so formed will help to regulate bandwidth, data traffic, in coding and decoding signals. Let A = (V(Λ), E(Λ)) beagraphwith|V(Λ)| = m and|V(Λ)| = n.Abijection from ζ : V(Λ)∪E(Λ) → {1, 2, 3, ⋯,m + n} is called (a, d)-edge antimagic total labeling if the edge-weights ζ(x) + ζ(xy) + ζ(y) for each xy ∈ E(Λ) form a sequence of consecutive positive integers with minimum edge-weight a and common difference d. In addition, it is called super (a, d)-edge antimagic total labeling if vertices receive the smallest labels. Enomoto et al. (2000) proposed the conjecture that every tree admits super (a, 0)-EAT labeling. In this note, bounds of the minimum and maximum edge-weights for super (a, d)-EAT labeling on the more generalized class of subdivided caterpillars are obtained. Moreover, we have investigated the existence of super (a, d)EAT labeling for the validation of the obtained bounds and the partial support of the aforesaid conjecture, where d ∈ {0, 1, 2}. In fact, the obtained results are a general extension of the results Akhlaq et al. [Utiltas Mathematica, 98 (2015), 227-249].

The use of numerical numbers to represent molecular networks plays a crucial role in the study of physicochemical and structural properties of the chemical compounds. For some integer k and a network G , the networks S k G and R k G are its derived networks called as generalized subdivided and generalized semitotal point networks, where S k and R k are generalized subdivision and generalized semitotal point operations, respectively. Moreover, for two connected networks, G 1 and G 2 , G 1 G 2 S k and G 1 G 2 R k are T -sum networks which are obtained by the lexicographic product of T G 1 and G 2 , respectively, where T ε S k , R k . In this paper, for the integral value k ≥ 1 , we find exact values of the first and second Zagreb indices for generalized T -sum networks. Furthermore, the obtained findings are general extensions of some known results for only k = 1 . At the end, a comparison among the different generalized T -sum networks with respect to first and second Zagreb indices is also included.

Let Γ = V , E be a graph. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral graph. A graph Γ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph Γ = Cay ℤ n , S , where n = p m ( p is a prime integer and m ∈ ℕ ) and S = a ∈ ℤ n | a , n = 1 . First, we show that Γ is an integral graph. Also, we determine the automorphism group of Γ . Moreover, we show that Γ and K v ▽ Γ are determined by their spectrum.

A molecular descriptor is a mathematical measure that associates a molecular graph with some real numbers and predicts the various biological, chemical, and structural properties of the underlying molecular graph. Wiener (1947) and Trinjastic and Gutman (1972) used molecular descriptors to find the boiling point of paraffin and total π -electron energy of the molecules, respectively. For molecular graphs, the general sum-connectivity and general Randić are well-studied fundamental topological indices (TIs) which are considered as degree-based molecular descriptors. In this paper, we obtain the bounds of the aforesaid TIs for the generalized F -sum graphs. The foresaid TIs are also obtained for some particular classes of the generalized F -sum graphs as the consequences of the obtained results. At the end, 3 D -graphical presentations are also included to illustrate the results for better understanding.

Topological index (TI) is a function that assigns a numeric value to a (molecular) graph that predicts its various physical and structural properties. In this paper, we study the sum graphs (S-sum, R-sum, Q-sum and T-sum) using the subdivision related operations and strong product of graphs which create hexagonal chains isomorphic to many chemical compounds. Mainly, the exact values of first general Zagreb index (FGZI) for four sum graphs are obtained. At the end, FGZI of the two particular families of sum graphs are also computed as applications of the main results. Moreover, the dominating role of the FGZI among these sum graphs is also shown using the numerical values and their graphical presentations.

A topological index is a numeric quantity assigned to a graph that characterizes the structure of a graph. Topological indices and physico-chemical properties such as atom-bond connectivity ABC , Randić, and geometric-arithmetic index GA are of great importance in the QSAR/QSPR analysis and are used to estimate the networks. In this area of research, graph theory has been found of considerable use. In this paper, the distinct degrees and degree sums of enhanced Mesh network, triangular Mesh network, star of silicate network, and rhenium trioxide lattice are listed. The edge partitions of these families of networks are tabled which depend on the sum of degrees of end vertices and the sum of the degree-based edges. Utilizing these edge partitions, the closed formulae for some degree-based topological indices of the networks are deduced.

In this paper, we give the relation between the spectrum of strongly regular graph and its clique-inserted graph. The Laplacian spectrum and the signless Laplacian spectrum of clique-inserted graph of strongly regular graph are calculated. We also give formulae expressing the energy, Kirchoff index, and the number of spanning trees of clique-inserted graph of a strongly regular graph. And, clique-inserted graph of the triangular graph T t , which is a strongly regular graph, is enumerated.

Let G = V E , E G be a connected graph with vertex set V G and edge set E G . For a graph G, the graphs S(G), R(G), Q(G), and T(G) are obtained by applying the four subdivisions related operations S, R, Q, and T, respectively. Further, for two connected graphs G 1 and G 2 , G 1 + F G 2 are F -sum graphs which are constructed with the help of Cartesian product of F G 1 and G 2 , where F ∈ S , R , Q , T . In this paper, we compute the lower and upper bounds for the first Zagreb coindex of these F -sum (S-sum, R-sum, Q-sum, and T-sum) graphs in the form of the first Zagreb indices and coincides of their basic graphs. At the end, we use linear regression modeling to find the best correlation among the obtained results for the thirteen physicochemical properties of the molecular structures such as boiling point, density, heat capacity at constant pressure, entropy, heat capacity at constant time, enthalpy of vaporization, acentric factor, standard enthalpy of vaporization, enthalpy of formation, octanol-water partition coefficient, standard enthalpy of formation, total surface area, and molar volume.

A connected graph is called Hamilton-connected if there exists a Hamiltonian path between any pair of its vertices. Determining whether a graph is Hamilton-connected is an NP-complete problem. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering. The detour index of a graph is defined to be the sum of lengths of detours between all the unordered pairs of vertices. The detour index has diverse applications in chemistry. Computing the detour index for a graph is also an NP-complete problem. In this paper, we study the Hamilton-connectivity of convex polytopes. We construct three infinite families of convex polytopes and show that they are Hamilton-connected. An infinite family of non-Hamilton-connected convex polytopes is also constructed, which, in turn, shows that not all convex polytopes are Hamilton-connected. By using Hamilton connectivity of these families of graphs, we compute exact analytical formulas of their detour index.

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

A numeric parameter which studies the behaviour, structural, toxicological, experimental, and physicochemical properties of chemical compounds under several graphs’ isomorphism is known as topological index. In 2018, Ali and Trinajstić studied the first Zagreb connection index Z C 1 to evaluate the value of a molecule. This concept was first studied by Gutman and Trinajstić in 1972 to find the solution of π -electron energy of alternant hydrocarbons. In this paper, the first Zagreb connection index and coindex are obtained in the form of exact formulae and upper bounds for the resultant graphs in terms of different indices of their factor graphs, where the resultant graphs are obtained by the product-related operations on graphs such as tensor product, strong product, symmetric difference, and disjunction. At the end, an analysis of the obtained results for the first Zagreb connection index and coindex on the aforesaid resultant graphs is interpreted with the help of numerical values and graphical depictions.

Let G be a simple connected graph. Suppose Δ = Δ 1 , Δ 2 , … , Δ l an l -partition of V G . A partition representation of a vertex α w . r . t Δ is the l − vector d α , Δ 1 , d α , Δ 2 , … , d α , Δ l , denoted by r α | Δ . Any partition Δ is referred as resolving partition if ∀ α i ≠ α j ∈ V G such that r α i | Δ ≠ r α j | Δ . The smallest integer l is referred as the partition dimension pd G of G if the l -partition Δ is a resolving partition. In this article, we discuss the partition dimension of kayak paddle graph, cycle graph with chord, and a graph generated by chain of cycles. It has been shown that the partition dimension of the said families of graphs is constant.

Let A ( G ) be the adjacent matrix and D ( G ) the diagonal matrix of the degrees of a graph G, respectively. For 0 &le, &alpha, &le, 1 , the A &alpha, matrix is the general adjacency and signless Laplacian spectral matrix having the form of A &alpha, ( G ) = &alpha, D ( G ) + ( 1 &minus, ) A ( G ) . Clearly, A 0 ( G ) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A &alpha, spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results.

Let G be a finite, connected graph of order of, at least, 2 with vertex set VG and edge set EG. A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S. The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψG. In this paper, first, we construct a class of graphs of order 2n+Σr=1k−2nmr, denoted by LSGn,m,k, and call these graphs as the layer Sun graphs with parameters n, m, and k. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSGn,m,k and the line graph of the layer Sun graph LSGn,m,k.

A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire π-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.

Labeling of graphs with numbers is being explored nowadays due to its diverse range of applications in the fields of civil, software, electrical, and network engineering. For example, in network engineering, any systems interconnected in a network can be converted into a graph and specific numeric labels assigned to the converted graph under certain rules help us in the regulation of data traffic, connectivity, and bandwidth as well as in coding/decoding of signals. Especially, both antimagic and magic graphs serve as models for surveillance or security systems in urban planning. In 1998, Enomoto et al. introduced the notion of super a,0 edge-antimagic labeling of graphs. In this article, we shall compute super a,0 edge-antimagic labeling of the rooted product of Pn and the complete bipartite graph K2,m combined with the union of path, copies of paths, and the star. We shall also compute a super a,0 edge-antimagic labeling of rooted product of Pn with a special type of pancyclic graphs. The labeling provided here will also serve as super a′,2 edge-antimagic labeling of the aforesaid graphs. All the structures discussed in this article are planar. Moreover, our findings have also been illustrated with examples and summarized in the form of a table and 3D plots.

Let Gbe a graph with n vertices and Γu be the degree of its u-th vertex (Γx is the degree of u). In this article, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si2C3−Ip,q, Si2C3−IIp,q, Si2C3−IIIp,q, and SiC3−IIIp,q.

The Resistance-Harary index of a connected graph G is defined as R H ( G ) = ∑ { u , v } ⊆ V ( G ) 1 r G ( u , v ) , where r G ( u , v ) is the resistance distance between vertices u and v in G . A connected graph G is said to be a cactus if each of its blocks is either an edge or a cycle. Let C n k be the set of all cacti of order n containing exactly k cycles. In this paper, we characterize the graphs with maximum Resistance-Harary index among all graphs in C n k .

The total domination number of a graph G without isolated vertices is the minimum number of vertices that dominate all vertices in G. The total bondage number of G is the minimum number of edges whose removal enlarges the total domination number. In this paper, we establish a tight lower bound for the total bondage number of a vertex-transitive graph. We also obtain upper bounds for regular graphs by investigating the relation between the total bondage number and the efficient total domination. As applications, we study the total bondage numbers for some circulant graphs and toroidal meshes by characterizing the existence of efficient total dominating sets in these graphs.

The resistance distance is widely used in random walk, electronic engineering, and complex networks. One of the main topics in the study of the resistance distance is the computation problem. The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$ . Two classes of new corona graphs, the corona-vertex of the subdivision graph $G_{1}\diamondsuit G_{2}$ and the corona-edge of the subdivision graph $G_{1}\star G_{2}$ , were defined by Lu and Miao. The adjacency spectrum and the signless Laplacian spectrum of the two new graph operations were computed when $G_{1}$ is an arbitrary graph and $G_{2}$ is an $r_{2}$ -regular graph. In this paper, we give the closed-form formulas for the resistance distance and Kirchhoff index of $G_{1}\diamondsuit G_{2}$ and $G_{1}\star G_{2}$ in terms of the resistance distance and Kirchhoff index of the factor graphs.

The atom-bond connectivity (ABC) index is one of the most actively studied degree-based graph invariants, which are found in a vast variety of chemical applications. For a simple graph $G$ , it is defined as $ABC(G)=\sum _{uv \in E(G)} ({({d(u)+d(v)-2})/({d(u) d(v)})})^{1/2}$ , where $d(v)$ denotes the degree of a vertex $v$ of $G$ . Recently in [17] graphs with $n$ vertices, $2n-4$ and $2n-3$ edges, and maximum $ABC$ index were characterized. Here, we consider the next, more complex case, and characterize the graphs with $n$ vertices, $2n-2$ edges, and maximum $ABC$ index.

An outer-independent total dominating set (OITDS) of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$ , and the set $V(G)\setminus D$ is independent. The outer-independent total domination number of a graph $G$ , denoted by $\gamma _{oit}(G)$ , is the minimum cardinality of an OITDS of $G$ . An outer-independent total Roman dominating function (OITRDF) on a graph $G$ is a function $f: V(G) \rightarrow \{0, 1, 2\}$ satisfying the conditions that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$ , every vertex $x$ with $f(x)\geq 1$ is adjacent to at least one vertex $y$ with $f(y)\geq 1$ , and any two different vertices $a,b$ with $f(a)=f(b)=0$ are not adjacent. The minimum weight $\omega (f) =\sum _{v\in V(G)}f(v)$ of any OITRDF $f$ for $G$ is the outer-independent total Roman domination number of $G$ , denoted by $\gamma _{oitR}(G)$ . A graph $G$ is called an outer-independent total Roman graph (OIT-Roman graph) if $\gamma _{oitR}(G)=2\gamma _{oit}(G)$ . In this paper, we propose dynamic programming algorithms to compute the outer-independent total domination number and the outer-independent total Roman domination number of a tree, respectively. Moreover, we characterize all OIT-Roman graphs and give a linear time algorithm for recognizing an OIT-Roman graph.

Given a complete multipartite graph G , we explicitly formulate the effective resistance distance between any two vertices of G , using methods from linear algebra. The Kirchhoff index and the degree Kirchhoff index are defined in terms of the effective resistance distances for any simple graph. In this paper, for G , the exact formulae for the two indices are provided. Along with these results, we obtain the optimal Kirchhoff index and the optimal degree Kirchhoff index.

The first multiplicative Zagreb index of a graph G is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore the trees in terms of given number of vertices of maximum degree. The maximum and minimum values of ∏ 1 ( G ) and ∏ 2 ( G ) of trees with arbitrary number of maximum degree are provided. In addition, the corresponding extremal graphs are characterized.

The first multiplicative Zagreb index of a graph G is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the degree of each edge over all edges. In our work, we explore the multiplicative Zagreb indices of bipartite graphs of order n with diameter d , and sharp upper bounds are obtained for these indices of graphs in B ( n , d ) , where B ( n , d ) is the set of all n -vertex bipartite graphs with the diameter d . In addition, we explore the relationship between the maximal multiplicative Zagreb indices of graphs within B ( n , d ) . As consequences, those bipartite graphs with the largest, second-largest and smallest multiplicative Zagreb indices are characterized, and our results extend and enrich some known conclusions.