204 Results Found

The vertex-degree function index $H_f\left(\mathsf{\Gamma }\right)$ is defined as $H_f\left(\mathsf{\Gamma }\right)={\sum }_{v\in V\left(\mathsf{\Gamma }\right)}f\left(d\left(v\right)\right)$ for a function $f\left(x\right)$ defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degr...

We find a lower bound for the k-subdomination number on the set of graphs with a given upper bound for vertex degrees. We study the cases where the proposed lower bound is sharp, construct the optimal graphs and indicate the corresponding k-subdomina...

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-de...

In this article, we introduce the notions of maximal products of fuzzy graph structures, regular fuzzy graph structures, and describe these notions with examples and properties. Further, we present the degree and total degree of a vertex in maximal p...

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices...

A complex fuzzy set (CFS) is described by a complex-valued truth membership function, which is a combination of a standard true membership function plus a phase term. In this paper, we extend the idea of a fuzzy graph (FG) to a complex fuzzy graph (C...

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 ru...

The reformulated Zagreb indices of a graph are obtained from the classical Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of the end vertices of the edge minus 2. In this paper...

In this paper, we investigate the existence of perfect $(1,2)$-dominating sets ($(1,2)$-PDS) in graphs with three vertices of maximum degree equal to $n-2$. A perfect $(1,2)$-dominating set is a special case of a $(1,2)$-dominating set. Graphs with such...

Let G be a connected (molecular) graph with the vertex set $V\left(G\right)=\{v_1,\cdots ,v_n\}$, and let $d_i$ and ${\sigma }_i$ denote, respectively, the vertex degree and the transmission of $v_i$, for $1\le i\le n$. In this paper, we aim to provide a new matrix description of the celebrated Wi...

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 ${\mathcal{I}}_{\mathfrak{f}}($...

In this paper, we present a methodological framework for conceptual modeling of assembly supply chain (ASC) networks. Models of such ASC networks are divided into classes on the basis of the numbers of initial suppliers. We provide a brief overview o...

Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree ( $DD$ ) index was introduced...

A novel topological index, the face index ( $FI$ ), is proposed in this paper. For a molecular graph G, face index is defined as $FI\left(G\right)={\sum }_{f\in F\left(G\right)}d\left(f\right)={\sum }_{v\sim f,f\in F\left(G\right)}d\left(v\right)$ , where...

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 th...

Minimal Strong Digraphs (MSDs) can be regarded as a generalization of the concept of tree to directed graphs. Their cyclic structure and some spectral properties have been studied in several articles. In this work, we further study some properties of...

In this paper, we study the class of graphs $G_{m,n}$ that have the same degree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class ${\overline{G}}_{m,n}$ of the complements of such graphs. The problems of finding a maximum clique and a maximum independe...

Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let $G=\left(V\right(G),E(G\left)\right)$ be a graph. Then, the Sombor index of G is defined as $SO\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\sqrt{d_G^2\left(u\right)+d_G^2\left(v\right)}$. In this paper, we give some lemmas that c...

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...

The importance of individuals and groups in networks is modeled by various centrality measures. Additionally, Freeman’s centralization is a way to normalize any given centrality or group centrality measure, which enables us to compare individua...

In this work, we investigate essential definitions, defining $G$ as a simple graph with vertices in ${\mathbb{Z}}_n$ and subgraphs ${\Gamma }_u$ and ${\Gamma }_q$ as unit residue and quadratic residue graphs modulo $n$, respectively. The investigation extends to the degree of $G$, ${\Gamma }_u$, and ${\Gamma }_q$...

Deployable structures based on origami are widely used in the application of actuators. In this paper, we present a novel family of origami-based deployable structures with constant curvature. Two categories of non-flat-foldable and non-developable d...

The study of networks and graphs carried out by topological measures performs a vital role in securing their hidden topologies. This strategy has been extremely used in biomedicine, cheminformatics and bioinformatics, where computations dependent on...

For a (molecular) graph G, the extended adjacency index $EA\left(G\right)$ is defined as Equation (1). In this paper we introduce some graph transformations which increase or decrease the extended adjacency ( $EA$ ) index. Also, we obtain the ext...

The rhenadicarbaborane carbonyl nitrosyls (C₂B_n−3H_n−1)Re(CO)₂(NO), (n = 8 to 12), of interest in drug delivery agents based on the experimentally known C₂B₉H₁₁Re(CO)₂(NO) and related species, have been investigated by density functional t...

This paper introduces a variant of entropy measures based on vertex eccentricity and applies it to all graphs representing the isomers of octane. Taking into account the vertex degree as well (degree-ecc-entropy), we find a good correlation with the...

In this paper, we consider the minimal vertex cover and minimal dominating sets with capacity and/or connectivity constraint enumeration problems. We develop polynomial-delay enumeration algorithms for these problems on bounded-degree graphs. For the...

The incidence of edges on vertices is a cornerstone of graph theory, with profound implications for various graph properties and applications. Understanding degree distributions and their implications is crucial for analyzing and modeling real-world...

The clustering coefficient of a vertex v, of degree at least 2, in a graph $\Gamma$ is obtained using the formula $C\left(v\right)=\frac{2t\left(v\right)}{\mathrm{deg}\left(v\right)\left(\mathrm{deg}\right(v)-1)},$ where $t\left(v\right)$ denotes the number of triangles of the graph containing v as a vertex, and the clustering co...

A Multinomial Processing Tree (MPT) is a directed tree with a probability associated with each arc and partitioned terminal vertices. We consider an additional parameter for each arc, a measure such as time. Each vertex represents a process. An arc d...

For a graph G, the resistance distance $r_G(x,y)$ is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index $R^{\ast }(G)={\sum }_{\{x,y\}\subset V(G)}{d}_G(x)$...

An efficient total dominating set D of a graph G is a vertex subset such that every vertex of G has exactly one neighbor in the set D. In this paper, we give necessary and sufficient conditions for the existence of efficient total domination sets of...

In the Vertex Cover Reconfiguration (VCR) problem, given a graph G, positive integers k and ℓ and two vertex covers S and T of G of size at most k, we determine whether S can be transformed into T by a sequence of at most ℓ vertex additions or remova...

By swapping out atoms for vertices and bonds for edges, a graph may be used to model any molecular structure. A graph G is considered to be a chemical graph in graph theory if no vertex of G has a degree of 5 or greater. The bond incident degree (BID...

The difference of Zagreb indices of a graph G is defined as $\Delta \mathrm{M}\left(G\right)={\displaystyle \sum _{u\in V\left(G\right)}}{\left(d\left(u\right)\right)}^2-{\displaystyle \sum _{uv\in E\left(G\right)}}d\left(u\right)d\left(v\right)$, where $d\left(x\right)$ denotes the degree of a vertex x in G. A Halin graph G is a graph that results from a plane tree T without ver...

In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for ${\eta }_0\le \eta <0$ among all n-vertex unicyclic graphs with a fixed maximum degree $\mathsf{\Delta }$ and the maximum and the second maximum general Randić index f...

The geometries and energetics of the n-vertex polyhedral dicobaltadithiaboranes and dicobaltadiselenaboranes Cp₂Co₂E₂B_n−4H_n−4 (E = S, Se; n = 8 to 12) have been investigated via the density functional theory. Most of the lowest-energy str...

A vertex coloring of a graph G is a mapping that allots colors to the vertices of G. Such a coloring is said to be a proper vertex coloring if two vertices joined by an edge receive different colors. The chromatic number $\chi (G)$ is the le...

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...

Classical link prediction methods mainly utilize vertex information and topological structure to predict missing links in networks. However, accessing vertex information in real-world networks, such as social networks, is still challenging. Moreover,...

Let $G=(V,E)$ be a finite simple graph with $p=\left|V\right|$ vertices and $q=\left|E\right|$ edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection f from $V\cup E$ to the consecutive integers $1,2,\dots ,p+q,$ with the property that, f...

Topological indices are numerical invariants derived from graph structures that are essential tools used in computational chemistry and biology for encoding molecular information. By exploiting the inherent symmetries of molecular graphs, we develop...

Let G be a graph with n vertices, let $A\left(G\right)$ be an adjacency matrix of G and let $P_A(G,\lambda )$ be the characteristic polynomial of $A\left(G\right)$. The adjacency spectrum of G consists of eigenvalues of $A\left(G\right)$. A graph G is said to be determined by its adjacency s...

Against the background of the construction of new power systems, power generation, transmission, distribution, and dispatching services are open to the outside world for interaction, and the accessibility of attack paths has been significantly enhanc...

By analyzing the coverage model of cameras, a surveillance camera network model based on road vertex coverage is proposed, and an optimized deployment method for cameras based on the Minimum Weighted Vertex Cover (MWVC) model is given. The greedy alg...

In this paper, a case study is conducted based on the real data obtained from the local Distribution System Operator (DSO) of electrical energy. The analyzed network represents connections and high-voltage switchgears of 110 kV. Selected graph parame...

The topology of an interconnection network can be modeled by a graph $G=\left(V\right(G),E(G\left)\right)$. The connectivity of graph G is a parameter used to measure the reliability of a corresponding network. The direct product is an important graph product. This paper ma...

This study explores the degree energy of fuzzy graphs to establish fundamental spectral bounds and characterize adjacency structures. We derive upper bounds on the sum of squared degree eigenvalues based on vertex degree distributions and formulate c...

This paper presents an efficient algorithm for matching subgraph queries in a multi-graph based on features-based indexing techniques. The KD-tree data structure represents these nodes’ features, while the set-trie index data structure represen...

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 m...