Search Results (16)

Fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) that is used to describe the topological structural complexity or degree of disorder in networks. Topological indices, as graph invariants, provide quantitative descriptors for characterizing global structural properties. In this paper, we investigate two types of generalized Sierpiński graphs constructed on the basis of different seed graphs, and employ six topological indices—the first Zagreb index, the second Zagreb index, the forgotten index, the augmented Zagreb index, the Sombor index, and the elliptic Sombor index—to analyze the corresponding entropy. We utilize the method of edge partition based on vertex degrees and derive analytical formulations for the first Zagreb entropy, the second Zagreb entropy, the forgotten entropy, the augmented Zagreb entropy, the Sombor entropy, and the elliptic Sombor entropy. This research approach, which integrates entropy with Sierpiński network characteristics, furnishes novel perspectives and instrumental tools for addressing challenges in chemical graph theory, computer networks, and other related fields. Full article

Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows: $e^{\mathcal{ABC}}=e^{\mathcal{ABC}}({\rm Y})={\displaystyle \sum _{v_i{v}_j\in E({\rm Y})}}{e}^{\sqrt{{\displaystyle \frac{d_i+d_j-2}{d_i{d}_j}}}},$ where $d_i$ is the degree of the vertex $v_i$ in ${\rm Y}$. In this paper, we prove that the double star $D{S}_{n-3,1}$ is the second maximal graph with respect to the $e^{\mathcal{ABC}}$ index of trees of order n. We give an upper bound on $e^{\mathcal{ABC}}$ of unicyclic graphs of order n and characterize the maximal graphs. The graph $K_1\vee (P_3\cup (n-4)K_1)$ gives the maximal graph with respect to the $e^{\mathcal{ABC}}$ index of bicyclic graphs of order n. We present several relations between $e^{\mathcal{ABC}}({\rm Y})$ and $ABC({\rm Y})$ of graph ${\rm Y}$. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research. Full article

Consider a unicyclic graph G with edge set $E\left(G\right)$. Let $\mathfrak{f}$ be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G is defined as ${\mathcal{I}}_{\mathfrak{f}}\left(G\right)={\sum }_{xy\in E\left(G\right)}\mathfrak{f}(d_G\left(x\right),d_G\left(y\right))$, where $d_G\left(x\right)$ denotes the degree a vertex x in G. This paper determines optimal bounds for ${\mathcal{I}}_{\mathfrak{f}}\left(G\right)$ in terms of the order of G and a parameter $\mathfrak{z}$, where $\mathfrak{z}$ is either the number of pendent vertices of G or the matching number of G. The paper also fully characterizes all unicyclic graphs that achieve these bounds. The function $\mathfrak{f}$ must satisfy specific requirements, which are met by several popular indices, including the Sombor index (and its reduced version), arithmetic–geometric index, sigma index, and symmetric division degree index. Consequently, the general results obtained provide bounds for several well-known indices. Full article

A vertex-degree-based topological index $\phi$ associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in QSPR/QSAR investigations. A subset of k edges in G with no common vertices is called a k-matching of G, and the number of such subsets is denoted by $m\left(G,k\right)$. Recently, this number was naturally extended to weighted graphs, where the weight function is induced by the topological index $\phi$. This number was denoted by $m_k\left(G,\phi \right)$ and called the k-matchings of G with respect to the topological index $\phi$. It turns out that $m_1\left(G,\phi \right)=\phi \left(G\right),$ and so for $k\ge 2,$ the k-matching numbers $m_k\left(G,\phi \right)$ can be viewed as kth order topological indices which involve both the topological index $\phi$ and the k-matching numbers. In this work, we solve the extremal value problem for the number of 2-matchings with respect to general sum-connectivity indices ${\mathcal{SC}}_{\mathit{\alpha }}$, over the set ${\mathcal{T}}_n$ of trees with n vertices, when $\alpha$ is a real number in the interval $\left[-1,0\right).$ Full article

Vertex-degree-based (VDB) topological indices have been applied in the study of molecular structures and chemical properties. At present, the exponential VDB index has been studied extensively. Naturally, we began to consider the logarithmic VDB index $ln{\mathcal{T}}_f$. In this paper, we first discuss the necessity of a logarithmic VDB index, and then present sufficient conditions so that $P_n$ and $S_n$ are the only trees with the smallest and greatest values of $ln{\mathcal{T}}_f\left(T\right)$. As applications, the minimal and maximal trees of some logarithmic VDB indices are determined. Through our work, we found that the logarithmic VDB index $ln{\mathcal{T}}_f$ has excellent discriminability, but the relevant results are not completely opposite to the exponential VDB index. The study of logarithmic VDB indices is an interesting but difficult task that requires further resolution. Full article

A VDB (vertex-degree-based) topological index over a set of digraphs $\mathcal{H}$ is a function $\phi :\mathcal{H}\to \mathbb{R}$, defined for each $H\in \mathcal{H}$ as $$\phi \left(H\right)=\frac{1}{2}\sum _{uv\in E}{\phi }_{d_u^{+}{d}_v^{-}},$$ where E is the arc set of H, $d_u^{+}$ and $d_v^{-}$ denote the out-degree and in-degree of vertices u and v respectively, and ${\phi }_{ij}=f(i,j)$ for an appropriate real symmetric bivariate function f. It is our goal in this article to introduce a new approach where we base the concept of VDB topological index on the space of real matrices instead of the space of symmetric real functions of two variables. We represent a digraph H by the $p\times p$ matrix $\alpha \left(H\right)$, where ${\left[\alpha \left(H\right)\right]}_{ij}$ is the number of arcs $uv$ such that $d_u^{+}=i$ and $d_v^{-}=j$, and p is the maximum value of the in-degrees and out-degrees of H. By fixing a $p\times p$ matrix $\phi$, a VDB topological index of H is defined as the trace of the matrix ${\phi }^T\alpha \left(H\right)$. We show that this definition coincides with the previous one when $\phi$ is a symmetric matrix. This approach allows considering nonsymmetric matrices, which extends the concept of a VDB topological index to nonsymmetric bivariate functions. Full article

A vertex degree based topological index called the Sombor index was recently defined in 2021 by Gutman and has been very popular amongst chemists and mathematicians. We determine the amount of change of the Sombor index when some elements are removed from a graph. This is done for several graph elements, including a vertex, an edge, a cut vertex, a pendant edge, a pendant path, and a bridge in a simple graph. Also, pendant and non-pendant cases are studied. Using the obtained formulae successively, one can find the Sombor index of a large graph by means of the Sombor indices of smaller graphs that are just graphs obtained after removal of some vertices or edges. Sometimes, using iteration, one can manage to obtain a property of a really large graph in terms of the same property of many other subgraphs. Here, the calculations are made for a pendant and non-pendant vertex, a pendant and non-pendant edge, a pendant path, a bridge, a bridge path from a simple graph, and, finally, for a loop and a multiple edge from a non-simple graph. Using these results, the Sombor index of cyclic graphs and tadpole graphs are obtained. Finally, some Nordhaus–Gaddum type results are obtained for the Sombor index. Full article

The importance of symmetry in graph theory has always been significant, but in recent years, it has become much more so in a number of subfields, including but not limited to domination theory, topological indices, Gromov hyperbolic graphs, and the metric dimension of graphs. The purpose of this monograph is to initiate the idea of a multi polar q-rung orthopair fuzzy graphs (m-$\mathfrak{PqROPFG})$ as a fusion of multi polar fuzzy graphs and q-rung orthopair fuzzy graphs. Moreover, for a vertex of multi polar q-rung orthopair fuzzy graphs, the degree and total degree of the vertex are defined. Then, some product operations, inclusive of direct, Cartesian, semi strong, strong lexicographic products, and the union of multi polar q-rung orthopair fuzzy graphs (m-$\mathfrak{PqROPFG}s)$, are obtained. Also, at first we define some degree based fuzzy topological indices of m-$\mathfrak{PqROPFG}$. Then, we compute Zareb indices of the first and second kind, Randic indices, and harmonic index of a m-$\mathfrak{PqROPFG}$. Full article

Graph theory serves as an engaging arena for the investigation of proof methods within the field of discrete mathematics, and its findings find practical utility in numerous scientific domains. Chemical graph theory is a specialized branch of mathematics that uses graphs to represent and analyze the structure and properties of chemical compounds. Topological indices are mathematical properties of graphs that play a crucial role in chemistry. They provide a unique way to connect the structural characteristics of chemical compounds to their corresponding molecular graphs. The flabellum graph $F_n^{(k,j)}$ is obtained with the help of $k\ge 2$ duplicates of the cycle graph $C_n$ with a common vertex (known as, central vertex). Then, in j of these duplicates, additional edges are added, joining the central vertex to all non-adjacent vertices. In this article, we compute different degree-based topological indices for flabellum graphs, including some well known indices, such as the Randić index, the atom bond connectivity index, the geometric–arithmetic index, and the Zagreb indices. This research provides an in-depth examination of these specific indices within the context of flabellum graphs. Moreover, the behavior of these indices is shown graphically, in terms of the parameters $j,k$, and n. Additionally, we have extended the concept of the first Zagreb index, to address the issue of cybercrime. This application enables us to identify criminals who exhibit higher levels of activity and engagement in multiple criminal activities when compared to their counterparts. Furthermore, we conducted a comprehensive comparative analysis of the first Zagreb index against the closeness centrality measure. This analysis sheds light on the effectiveness and relevance of the topological index in the context of cybercrime detection and network analysis. Full article

Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant $GR{M}_{\alpha }$, known under the name general reduced second Zagreb index, is defined as $GR{M}_{\alpha }\left(\mathrm{\Gamma }\right)={\sum }_{uv\in E\left(\mathrm{\Gamma }\right)}(d_{\mathrm{\Gamma }}\left(u\right)+\alpha )(d_{\mathrm{\Gamma }}\left(v\right)+\alpha )$, where $d_{\mathrm{\Gamma }}\left(v\right)$ is the degree of the vertex v of the graph $\mathrm{\Gamma }$ and $\alpha$ is any real number. In this paper, among all trees of order n, and all unicyclic graphs of order n with girth g, we characterize the extremal graphs with respect to $GR{M}_{\alpha }$$(\alpha \ge -\frac{1}{2})$. Using the extremal unicyclic graphs, we obtain a lower bound on $GR{M}_{\alpha }\left(\mathrm{\Gamma }\right)$ of graphs in terms of order n with k cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on $GR{M}_{\alpha }$ of different classes of graphs in terms of order n, size m, independence number $\gamma$, chromatic number k, etc. In particular, we present an upper bound on $GR{M}_{\alpha }$ of connected triangle-free graph of order $n>2$, $m>0$ edges with $\alpha >-1.5$, and characterize the extremal graphs. Finally, we prove that the Turán graph $T_n\left(k\right)$ gives the maximum $GR{M}_{\alpha }(\alpha \ge -1)$ among all graphs of order n with chromatic number k. Full article

A vertex-degree-based (VDB, for short) topological index $\phi$ induced by the numbers $\left\{{\phi }_{ij}\right\}$ was recently defined for a digraph D, as ${\displaystyle \phi \left(D\right)=\frac{1}{2}\sum _{uv}{\phi }_{d_u^{+}{d}_v^{-}},}$ where $d_u^{+}$ denotes the out-degree of the vertex $u,$$d_v^{-}$ denotes the in-degree of the vertex $v,$ and the sum runs over the set of arcs $uv$ of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over ${\mathcal{D}}_n$, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over $\mathcal{OT}\left(n\right)$ and $\mathcal{O}\left(G\right)$, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively. Full article

The concept of arithmetic-geometric index was recently introduced in chemical graph theory, but it has proven to be useful from both a theoretical and practical point of view. The aim of this paper is to obtain new bounds of the arithmetic-geometric index and characterize the extremal graphs with respect to them. Several bounds are based on other indices, such as the second variable Zagreb index or the general atom-bond connectivity index), and some of them involve some parameters, such as the number of edges, the maximum degree, or the minimum degree of the graph. In most bounds, the graphs for which equality is attained are regular or biregular, or star graphs. Full article

The concept of Sombor index $\left(SO\right)$ was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index $SO$: $SO=SO\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}\sqrt{d_G{\left(v_i\right)}^2+d_G{\left(v_j\right)}^2},$ where $d_G\left(v_i\right)$ is the degree of vertex $v_i$ in G. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work. Full article

In the studies of quantitative structure–activity relationships (QSARs) and quantitative structure–property relationships (QSPRs), graph invariants are used to estimate the biological activities and properties of chemical compounds. In these studies, degree-based topological indices have a significant place among the other descriptors because of the ease of generation and the speed with which these computations can be accomplished. In this paper, we give the results related to the first, second, and third Zagreb indices, forgotten index, hyper Zagreb index, reduced first and second Zagreb indices, multiplicative Zagreb indices, redefined version of Zagreb indices, first reformulated Zagreb index, harmonic index, atom-bond connectivity index, geometric-arithmetic index, and reduced reciprocal Randić index of a new graph operation named as “subdivision vertex-edge join” of three graphs. Full article

A graph G is called an ℓ-apex tree if there exist a vertex subset $A\subset V\left(G\right)$ with cardinality ℓ such that $G-A$ is a tree and there is no other subset of smaller cardinality with this property. In the paper, we investigate extremal values of several monotonic distance-based topological indices for this class of graphs, namely generalized Wiener index, and consequently for the Wiener index and the Harary index, and also for some newer indices as connective eccentricity index, generalized degree distance, and others. For the one extreme value we obtain that the extremal graph is a join of a tree and a clique. Regarding the other extreme value, which turns out to be a harder problem, we obtain results for $\ell =1$ and pose some open questions for higher ℓ. Symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including topological indices of graphs. Full article