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We offer an alternative proof, using the Stein-Chen method, of Bollobás' theorem concerning the distribution of the extreme degrees of a random graph. Our proof also provides a rate of convergence of the extreme degree to its asymptotic distribution. The same method also applies in a more general setting where the probability of every pair of vertices being connected by edges depends on the number of vertices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks. [ABSTRACT FROM AUTHOR]

Recently, finding extremal structures of graphs on Sombor index has received a lot of attention. The Sombor (SO) index of a graph G is defined by the sum of weights d e g G (u) 2 + d e g G (v) 2 over all edges uv of G, where d e g G (u) stands for the degree of vertex u in G. In this article, we obtain a lower bound on Sombor index of trees with a given order and total domination number, and characterize the trees achieving the bound. [ABSTRACT FROM AUTHOR]

A topological index is a numerical property of a molecular graph that explains structural features of molecules. The potential of topological indices to discriminate between distinct structures is a significant topic to investigate. In this context, the exponential degree-based indices were put forward in the literature. The present work focuses on the exponential augmented Zagreb index (EAZ), which is defined for a graph G as E A Z (G) = ∑ v i v j ∈ E (G) e d i d j d i + d j - 2 3 , where d i represents the degree of the vertex v i and E(G) denotes the edge set of G. This work characterizes the maximal unicyclic graph for EAZ in terms of graph order, which was posed as an open problem in the recent article Cruz et al. (MATCH Commun Math Comput Chem 88:481-503, 2022). [ABSTRACT FROM AUTHOR]

We refine the enumeration of fusene chains of a given length with respect to the number of turns by constructing bijections between such chains and ternary words. Explicit formulas thus obtained are then used to compute the expected values for the whole class of bond-additive degree-based topological indices over all such chains of a given length. The results are also applicable to several other classes of chemically interesting polycyclic chains, such as, e.g., phenylene and spiro chains. [ABSTRACT FROM AUTHOR]

The Sombor index is a novel vertex-degree-based topological index which was proposed by Gutman in 2021. This index has been shown to be helpful in predicting the enthalpy of vaporization and entropy of octane isomers. In 2022, Gutman put forward a new variant of the Sombor index and gave its geometric interpretation. This invariant which we call the geometric Sombor index can be considered as a practical tool for measuring irregularity in graphs. Our aim is to study some basic mathematical properties of this new Sombor-type invariant. Especially, we prove that for any tree (resp. unicyclic graph) with a fixed order and maximum degree Δ , the geometric Sombor index is bounded below by Δ 3 - Δ 2 (resp. Δ 3 - Δ - 6 2 ). Also the extremal trees and unicyclic graphs that achieve the lower bound are characterized. [ABSTRACT FROM AUTHOR]

Let G be a graph with V(G) and E(G), as vertex set and edge set, respectively. The atom-bond sum-connectivity (ABS) index is a vertex-based topological index which is defined as [Formula: see text] where [Formula: see text] is the degree of the vertex a. In this paper, we obtain sharp upper bounds for the ABS index of molecular trees in terms of order and number of branching vertices and vertices of degree two.

Topological indices are molecular descriptors associated with the physical, chemical, and biological properties of the molecules. The main goal of this paper is to explore some mathematical properties of generalized irregular neighborhood related descriptor. Also, the particular cases of these descriptors are calculated for the molecular structure of some basic polycyclic aromatic hydrocarbons (PAHs) along with statistical analysis. [ABSTRACT FROM AUTHOR]

For any connected graph G = (V , E) and any function f on the positive integers set Z + , vertex-degree function index H f (G) is defined as the sum of f (d G (v)) over v ∈ V , where d G (v) is the degree of v in G. For any n , k ∈ Z + with n ≥ k , connected graphs with n vertices and clique number k form the set W n , k . In this paper, for any strictly concave and increasing function f on Z + , we determine the maximal and minimal values of H f (G) over G ∈ W n , k , and characterize the corresponding graphs G ∈ W n , k with the extremal values. We also get the maximum vertex-degree function index H f (G) , where f(x) is a strictly concave and decreasing function for x ≥ 1 . [ABSTRACT FROM AUTHOR]

Let G be a simple graph with vertex set V = { v 1 , v 2 , … , v n } . The notion i ∼ j is used to indicate that the vertices v i and v j of G are adjacent. For a vertex v i ∈ V , let d i be the degree of v i . The harmonic-arithmetic (HA) index of G is defined as H A (G) = ∑ i ∼ j 4 d i d j (d i + d j) - 2 . In this paper, a considerable number of inequalities involving the HA index and other topological indices are derived. For every obtained inequality, all the graphs that satisfy the equality case are also characterized. [ABSTRACT FROM AUTHOR]

AbstractA classic result in Euclidean geometry asserts that every non-collinear set of n points in the Euclidean plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with an appropriate definition of line. This conjecture remains open even for graph metrics. In this paper, sufficient conditions on the girth to guarantee that a graph satisfies the conjecture are stated. We first study the existence of a universal line generated by an edge. Then, we focus on graphs of order n and maximum girth g with respect to their radius r. We prove that graphs with minimum degree δ ≥ 4 and girth g = 2r+1 or g = 2r have at least n distinct lines and we also partially solve the case when δ = 3. Graphs with cut-vertices are also studied, providing several upper bounds on the diameter, in terms of the girth, in order to assure that the graph satisfies the Chen-Chvátal conjecture. Finally, we prove that any triangle-free graph of order n, diameter 3 and minimum degree δ ≥ 4 has more than n distinct lines and we also partially solve the case when δ = 3. [ABSTRACT FROM AUTHOR]

In this paper, we consider the friendship paradox in the context of random walks and paths. Among our results, we give an equality connecting long-range degree correlation, degree variability, and the degree-wise effect of additional steps for a random walk on a graph. Random paths are also considered, as well as applications to acquaintance sampling in the context of core-periphery structure. [ABSTRACT FROM AUTHOR]

Graph-based topological index of a chemical compound has been instrumental in determining its physico-chemical properties. Over the years, this mathematical tool has proved to be a time and cost efficient aid for chemists, which enables the study of different chemicals sans laborious efforts. One of the recent researches in topological indices is the degree based index called Sombor index. Sombor index along with its variants have become a hotspot in the field of mathematical and chemical research. Several chemical compounds are being studied to identify their potential application. In this regard, carbon nanotubes are widely being analysed as they seem to be quite promising compounds with their strong electrical and structural properties. In this study, we compute the Sombor index, reduced Sombor index and average Sombor index of armchair and zigzag carbon nanotubes. Further, a comparison of armchair and zigzag carbon nanotubes is made on the basis of their topological index based entropies. [ABSTRACT FROM AUTHOR]

Using fuzzy graph theoretical and computational methods, the multidisciplinary discipline of chemical graph theory examines the molecular formation of a chemical molecule as a fuzzy graph and looks into related mathematical questions. An effective tool in this field that links a numerical value to a graph formation is the topological index. The neighborhood inverse sum indeg index stands out as a contemporary index derived from neighborhood degree considerations. Neighborhood inverse sum indeg index is a crucial graph metric that is applied in real-world applications like corporate networking and signalling on traffic. So this index is generalized in fuzzy graph. In this paper, the neighborhood inverse sum indeg index is explored for some fuzzy graphs like a tree, double star, subgraph, star, complete fuzzy graph, complete bipartite graph, etc.. We evaluate the effect of neighborhood inverse sum indeg index value when the vertex or edge is deleted from a graph. Moreover, the relationship between two isomorphic fuzzy graphs are determined. We have determined the boundedness of this index for some fuzzy graphs like C S (n , α 1) , K p 1 ∨ (K q 1 ∪ K r 1) , etc. Also a couple of results are found. We all aware that one cause of global warming is the carbon emission. Finally we have found out the country which needs to reduce the carbon emissions by applying the concept of neighborhood inverse sum indeg index. [ABSTRACT FROM AUTHOR]

Let G = {Gi: i ∈ [n]} be a collection of not necessarily distinct n-vertex graphs with the same vertex set V, where G can be seen as an edge-colored (multi)graph and each Gi is the set of edges with color i. A graph F on V is called rainbow if any two edges of F come from different Gis'. We say that G is rainbow pancyclic if there is a rainbow cycle Cℓ of length ℓ in G for each integer ℓ ∈ [3, n]. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem: If δ (G i) ≥ n 2 for each i ∈ [n], then there is a rainbow Hamiltonian cycle in G. In this paper, under the same condition, we show that G is rainbow pancyclic except that n is even and G consists of n copies of K n 2 , n 2 . This result supports the famous meta-conjecture posed by Bondy. [ABSTRACT FROM AUTHOR]

The atom-bond sum-connectivity (ABS) index of a graph GG with edges e1,…,em{e}_{1},\ldots ,{e}_{m} is the sum of the numbers 1−2(dei+2)−1\sqrt{1-2{\left({d}_{{e}_{i}}+2)}^{-1}} over 1≤i≤m1\le i\le m, where dei{d}_{{e}_{i}} is the number of edges adjacent to ei{e}_{i}. In this article, we study the maximum values of the ABS index over graphs with given parameters. More specifically, we determine the maximum ABS index of connected graphs of a given order with a fixed (i) minimum degree, (ii) maximum degree, (iii) chromatic number, (iv) independence number, or (v) number of pendent vertices. We also characterize the graphs attaining the maximum ABS values in all of these classes.

A sequence σ of p non-negative integers is unigraphic if it is the degree sequence of exactly one graph, up to isomorphism. A polyhedral graph is a 3-connected, planar graph. We investigate which sequences are unigraphic with respect to the class of polyhedral graphs, meaning that they admit exactly one realisation as a polyhedron. We focus on the case of sequences with largest entry p - 2 . We give a classification of polyhedral unigraphic sequences starting with p - 2 , p - 2 , as well as those starting with p - 2 and containing exactly one 3. Moreover, we characterise the unigraphic sequences where a few vertices are of high degree. We conclude with a few other examples of families of unigraphic polyhedra. [ABSTRACT FROM AUTHOR]

Analysis of the physical-chemical characteristics of chemical structures requires the incorporation of topological indices into account. A structure-based approach that concentrates antiviral treatment on the main protease of the SARS-COV-2 virus. One degree-based measure that is a little comparable to the Zagreb index and Lanzhou index. This paper analyzed the effectiveness of the chemical compounds pyroxene, amphibole, detour-saturated plants, and several antiviral medications listed for the Lanzhou index. The obtained results demonstrate a significant relationship between the Lanzhou index under investigation and the physicochemical features of potential antiviral medications. The Lanzhou index for antioxidant drugs for M polynomial is enumerated. [ABSTRACT FROM AUTHOR]

Topological indices are obtained from molecular graphs and are real numbers that can forecast the biological and physicochemical properties of several anti-cancer treatments, including skin cancer, breast cancer, and blood cancer. This article focuses on the application of topological indices in predicting the effectiveness of several drugs used to cure blood cancer, such as Pamidronic acid, Alpelisib, Prednisone, Olaparib, Ribociclib, Tucatinib, dexamethasone, docetaxel, Midostaurin, paclitaxel, toremifene, and venetoclax. The article investigates the mathematical relationships between physical and chemical qualities and data encoded in chemical structures under characteristics such as molecular weight, molar volume, and complexity. Several topological indices are used in this context to forecast the physicochemical characteristics of the drugs. [ABSTRACT FROM AUTHOR]

A topological index is a numerical descriptor in mathematical chemistry and graph theory that quantifies a molecule's structural properties without considering its three-dimensional arrangement. A crucial factor to consider when exploring topological indices is their capacity to distinguish between different structures. In light of this, the exponential vertex-degree-based topological index is put forward in the literature. The present work focuses on investigating the mathematical properties and application potential of the exponential augmented Zagreb index (EAZ). The EAZ index for a graph Υ is defined as E A Z (Υ) = ∑ v i v j ∈ E (Υ) e d i d j d i + d j - 2 3 , where d i represents the degree of a vertex v i . Crucial upper and lower bounds of EAZ for numerous classes of graphs like bipartite, unicyclic, bicyclic, chemical graph, and general graphs are derived. The bounds are computed in terms of different graph parameters including graph order, size, maximum degree, number of pendant vertices and independence number. The extremal graphs for which the bounds appear are also characterized. Moreover, the EAZ index is found to correlate well with some physico-chemical properties of octanes. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a simple graph with n vertices and m edges. ν(G) and c(G) = m − n + θ be the matching number and cyclomatic number of G, where θ is the number of connected components of G, respectively. Wang and Wong in [18] provided formulae for the upper and the lower bounds of the nullity η(G) of G as η(G) = n − 2ν(G) + 2c(G) and η(G) = n − 2ν(G) − c(G), respectively. In this paper, we restate the upper and the lower bounds of nullity η(G) of G utilizing omega invariant and inherently vertex degrees of G. Also, in the case of the maximal and the minimal nullity conditions are satisfied for G, we present both two main inequalities and many inequalities in terms of Omega invariant, analogously cyclomatic number, number of connected components and vertex degrees of G.

The present study investigates the complex topological characteristics of DNA networks, with a specific emphasis on the innovative metric known as Connection Number (CN) as a key factor in determining network structure. The Connection Number, represented as CN(v) for a vertex v, measures the count of unique paths that link v to every other vertex in the network. By employing rigorous mathematical modeling and analysis techniques, we are able to reveal the profound implications of CN (complex networks) in characterizing the structural robustness and dynamics of information flow within DNA networks. The study of how the theory of graphs and chemicals interact is known as chemical graph theory. This paper, computing the hyper Zagreb connection index, augmented connection index, inverse sum connection index, harmonic connection index, symmetric division connection index, geometric arithmetic connection index, and atom bond connectivity connection index, of two significant types of backbone DNA and Barycentric subdivision of backbone DNA networks. Direct method computation is used to produce these Connection-based topological descriptors. [ABSTRACT FROM AUTHOR]

Let G be a graph with V(G) and E(G), as vertex set and edge set respectively. The atom-bond sum-connectivity index is a degree-based topological index which is defined as A B S (G) = ∑ u v ∈ E (G) d u + d v - 2 d u + d v , where the degree of the vertex u is denoted by d u . In this article, our focus lies on investigating the maximum value of atom-bond sum-connectivity among the class of bicyclic graphs on n vertices. In addition, the role of atom-bond sum-connectivity in explaining structure–property relationship is also demonstrated. [ABSTRACT FROM AUTHOR]

Let G be a simple, connected graph of minimum degree δ and order n. If G is a C 4 -free graph and n ≤ 4 δ - 2 , we prove that G is Hamiltonian, G is the Petersen graph or G ∈ { K 1 , K 2 , 2 K 2 ∨ K 1 , K 2 ∨ K 1 ∨ K 1 ∨ K 2 } . Also, if G is a C 4 -free graph with n ≤ 4 δ - 1 , we prove that G is traceable or G = 3 K 2 ∨ K 1 . Given that G is a triangle-free graph with n ≤ 2 δ + 2 , we show that G is Hamiltonian or G ∈ { K 1 , K 2 , P 4 , G 2 , 6 , K δ , δ + 1 , K δ , δ + 2 } . We deduce that every graph G with n ≤ 2 δ + 1 is Hamiltonian or G ∈ { K 1 , K 2 , 2 K δ ∨ K 1 , K δ , δ + 1 + H } . If G has n ≤ 2 δ + 2 , we deduce that it is traceable or G = K δ , δ + 2 + H . A result by Yasemin et al. [On one extension of Dirac’s theorem on Hamiltonicity. Discrete. Appl. Math.252, 10–16 (2019)], is deduced. The results strengthens one of the results by Dirac [Some theorems on abstract graphs. Proc. Lond. Math. Soc.2, 69-81 (1952)] and classifies all non-traceable and non-Hamiltonian graphs of given order, apart from, providing sufficient conditions for traceability or Hamiltonicity. We consider simple graphs only. [ABSTRACT FROM AUTHOR]

Abstract Graph theory plays an important role in modeling, designing, and analyzing the structural characteristics of any molecular structure or complex network. The numerical quantities associated with a molecular structure are called topological descriptors. These descriptors are essential for understanding the topology of molecular structures and their physicochemical properties. In this paper, we study the molecular structure of crystal carbon graphite for r-level, which is one of the most well-known allotropes of carbon. Furthermore, some topological descriptors for the molecular structure of crystal carbon graphite have been computed using degree and neighborhood degree sum-based techniques. The obtained results may be used in understanding the structural characteristics and making predictions about certain physiochemical properties of crystal carbon graphite. [ABSTRACT FROM AUTHOR]

Topological indices are numerical molecular descriptors that reveals structure-property relationship of chemical structures without employing any wet lab. Among numerous kind of indices, neighborhood degree sum (NDS) based indices are recently found to be significant in explaining physico-chemical properties of molecules. The present work focuses on determining optimum values of NDS-indices for the class of silicate and oxide networks. Some general formulations of the indices are obtained first for dominating, regular triangulene, and rhombus type silicate and oxide networks, and Copper oxide network. Explicit expressions of some NDS-indices are then generated. Finally, extremal conditions for the indices are established in terms of different network parameters. A graphical comparison of the outcomes is also reported. [ABSTRACT FROM AUTHOR]

The atom-bond sum-connectivity index of a graph G is defined as A B S (G) = ∑ x y ∈ E (G) d x + d y - 2 d x + d y , where d x is the degree of the vertex x in G. In this paper, we study the ABS index of unicyclic graphs with a specified girth and determine the smallest, second smallest, third smallest, and fourth smallest ABS index. We also identify the graphs that achieve these extremal values. In addition, the ABS index is found to have significant potential in structure–property modelling for some compounds containing cyclic substructures. [ABSTRACT FROM AUTHOR]

For a graph G with vertex set V G and edge set E G , the symmetric division deg index is defined as SDD (G) = ∑ u v ∈ E G (d u d v + d v d u) , where d u denotes the degree of vertex u in G. In 2018, Furtula et al. confirmed the quality of SDD index exceeds that of some more popular VDB indices, in particular that of the GA index. They show a close connection between the SDD index and the earlier well-established GA index. Thus, it is meaningful and important to consider the chemical and mathematical properties of the SDD index. In this paper, we determine some sharp bounds on the symmetric division deg index of graphs and line graphs and characterize the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

Objective: A single valued neutrosophic set represented the uncertainty of real life situations in terms of membership (t) , indeterminacy (i) and non-membership (f) degree. However, this uncertainty cannot be limited to those three degrees; there is also an additional refusal degree. For this issue, the Turiyam set is an appropriate tool, which described the neutrosophic refusal degree of this situation as a liberal (l) degree in addition to those three degrees. The graphical representation of this situation is required for knowledge processing. For this purpose, the Turiyam graph was introduced as an extension of the single valued neutrosophic graph. This graph is helpful when the depictions of the vertices or their relationships or both, are considered in terms of membership (t) , indeterminacy (i) , non-membership (f) and liberal (l) degrees. The goal of this paper is to introduce the degree, order and size in the context of Turiyam graphs and examine a social network (SN) with the help of this graph. Results: In this regard, the degree, order and size in the context of Turiyam graphs are studied. The feasibility of this Turiyam graph is shown by employing its concept in a social network (SN). Finally, the advantage of the Turiyam graph over the existing graph theories is recognized by viewing its better framework. [ABSTRACT FROM AUTHOR]

Chemical graph theory is the field which deals with the combination of chemistry and graph theory. In this paper, we find the degree based first and second K Bbanhatti, first and second hyper K Banhatti, first and second multiplicative K banhatti, first and second multiplicative hyper K banhatti, Sombor, KG, modified KG, multiplicative KG, multiplicative modified KG, K Harmonic and multiplicative K Harmonic Banhatti, first Banhatti and reduced Banhatti Sombor, delta Banhatti Sombor indices for backbone DNA and subdivided backbone DNA networks. These topological descriptors are computed by direct method. [ABSTRACT FROM AUTHOR]

We present lower and upper bounds for the geometric–arithmetic index of unicyclic graphs and provide extremal graphs for the corresponding bounds. [ABSTRACT FROM AUTHOR]

Given a simple graph G, the irregularity strength of G, denoted by s(G), is the least positive integer k such that there is a weight assignment on edges f : E (G) → { 1 , 2 , ⋯ , k } attributing distinct weighted degrees: f ~ (v) : = ∑ u : { u , v } ∈ E (G) f ({ u , v }) to all vertices v ∈ V (G) . It is straightforward that s (G) ≥ n / d for every d-regular graph G on n vertices with d > 1 . In 1987, Faudree and Lehel conjectured in turn that there is an absolute constant c such that s (G) ≤ n / d + c for all such graphs. Even though the conjecture has remained open in almost all relevant cases, it is more generally believed that there exists a universal constant c such that s (G) ≤ n / δ + c for every graph G on n vertices with minimum degree δ ≥ 1 which does not contain an isolated edge; In this paper we confirm that the generalized Faudree–Lehel Conjecture holds for graphs with δ ≥ n β where β is any fixed constant larger than 0.8; Furthermore, we confirm that the conjecture holds in general asymptotically. That is, we prove that for any ε ∈ (0 , 0.25) there exist absolute constants c 1 , c 2 such that for all graphs G on n vertices with minimum degree δ ≥ 1 and without isolated edges, s (G) ≤ n δ (1 + c 1 δ ε) + c 2 ; We thereby extend in various aspects and strengthen a recent result of Przybyło, who showed that s (G) ≤ n d (1 + 1 ln ε / 19 n ) = n d (1 + o (1)) for d-regular graphs with d ∈ [ ln 1 + ε n , n / ln ε n ] . We also improve the earlier general upper bound: s (G) < 6 n δ + 6 of Kalkowski, Karoński and Pfender. [ABSTRACT FROM AUTHOR]

For 0 ≤ ℓ < k , a Hamilton ℓ -cycle in a k-uniform hypergraph H is a cyclic ordering of the vertices of H in which the edges are segments of length k and every two consecutive edges overlap in exactly ℓ vertices. We show that for all 0 ≤ ℓ < k - 1 , every k-graph with minimum co-degree δ n with δ > 1 / 2 has (asymptotically and up to a subexponential factor) at least as many Hamilton ℓ -cycles as a typical random k-graph with edge-probability δ . This significantly improves a recent result of Glock, Gould, Joos, Kühn and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values 0 ≤ ℓ < k - 1 . [ABSTRACT FROM AUTHOR]