Given a connected graph G=(V(G),E(G)), a set S⊆V(G) is said to be a k-metric generator for G if any pair of different vertices in V(G) is distinguished by at least k elements of S. A metric generator of minimum cardinality among all k-metric generato...
The Resistance-Harary index of a connected graph G is defined as R H ( G ) = ∑ { u , v } ⊆ V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a...
The cutwidth minimization problem consists of finding an arrangement of the vertices of a graph G on a line Pn with n=|V(G)| vertices in such a way that the maximum number of overlapping edges (i.e., the congestion) is minimized. A graph G with a cut...
Recently, the exponential arithmetic–geometric index (EAG) was introduced. The exponential arithmetic–geometric index (EAG) of a graph G is defined as EAG(G)=∑vivj∈E(G)edi+dj2didj, where di represents the degree of the vertex vi...
In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for η0≤η<0 among all n-vertex unicyclic graphs with a fixed maximum degree Δ and the maximum and the second maximum general Randić index f...
The Hosoya index of a graph is defined by the total number of the matchings of the graph. In this paper, we determine the maximum Hosoya index of unicyclic graphs with n vertices and diameter 3 or 4. Our results somewhat answer a question proposed by...
The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices...
Given a simple connected graph G, the metric dimension dim(G) (and edge metric dimension edim(G)) is defined as the cardinality of a smallest vertex subset S⊆V(G) for which every two distinct vertices (and edges) in G have distinct distances to...
In discrete mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Chemical graph theory is concerned with non-trivial applications of graph theory to the solution of mole...
A set S ⊆ V ( G ) in a graph G is a dominating set if S dominates all vertices in G, where we say a vertex dominates each vertex in its closed neighbourhood. A set is independent if it is pairwise non-adjacent. The minimum cardinality of...
A topological index being a graph theoretic parameter plays a role of function for the assignment of a numerical value to a molecular graph which predicts the several physical and chemical properties of the underlying molecular graph such as heat of...
In this paper, completely regular endomorphisms of unicyclic graphs are explored. Let G be a unicyclic graph and let c E n d ( G ) be the set of all completely regular endomorphisms of G. The necessary and sufficient conditions under which...
For a connected graph G with vertex set V, denote by d(v) the degree of vertex v and d(u, v) the distance between u and v. The value Gut(G)=∑{u,v}⊆Vd(u)d(v)d(u,v) is called the Gutman index of G. Recently, the graph minimizing the Gutman ind...
The symmetric division deg (SDD) index is among the 148 discrete Adriatic indices that were developed about a decade ago. Motivated by the success of the SDD index, Ghorbani et al. in a recent paper proposed the inverse version of this index and call...
For a connected graph G, let DL(G) be its distance Laplacian matrix and λ1(G)≥λ2(G)≥…≥λn−1(G)>λn(G)=0 be its DL eigenvalues. The DL energy of G is defined as DLEG=∑i=1nλi(G)−2WGn,...
For a graph, the SK index is equal to the half of the sum of the degrees of the vertices, the SK1 index is equal to the half of the product of the degrees of the vertices, and the SK2 index is equal to the half of the square of the sum of the degrees...
For a∈R, the general sum-connectivity index of a graph G is defined as χa(G)=∑uv∈E(G)[dG(u)+dG(v)]a, where E(G) is the set of edges of G and dG(u) and dG(v) are the degrees of vertices u and v, respectively. For trees and unicyclic...
Grey matter volume reductions in the right superior temporal gyrus (rSTG) were observed in young adults who learned to ride a unicycle. As these decreases were correlated with the acquired ability in unicycling, the authors interpreted the change as...
The field of indices has been explored and advanced by various researchers for different purposes. One purpose is the optimization of indices in various problems. In this work, the general power-sum connectivity index is considered. The general power...
The field related to indices was developed by researchers for various purposes. Optimization is one of the purposes used by researchers in different situations. In this article, a generalized Sombor index is considered. This work is related to the id...
The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize it...
Consider a unicyclic graph G with edge set E(G). Let 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 If(...
It is well-known that the necessary and sufficient condition for a connected graph to be unicyclic is that its omega invariant, a recently introduced graph invariant useful in combinatorial and topological calculations, is zero. This condition could...
This paper presents a comprehensive review of the literature on the original concept of metric location, along with its various adaptations and extensions that have been developed over time. Given that determining a minimum location set is generally...
A locally irregular graph is a graph in which the end vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A gra...
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, t...
Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let G=(V(G),E(G)) be a graph. Then, the Sombor index of G is defined as SO(G)=∑uv∈E(G)dG2(u)+dG2(v). In this paper, we give some lemmas that c...
Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph Ω provides the multiplicative first Zagreb index of a graph. In the context of graph theor...
The extremal problem of Laplacian permanents of graphs is a classical and challenging topic in algebraic combinatorics, where the inherent #P-complete complexity of permanent computation renders this pursuit particularly intractable. In this paper, w...
For a (molecular) graph G, the extended adjacency index E A ( G ) is defined as Equation (1). In this paper we introduce some graph transformations which increase or decrease the extended adjacency ( E A ) index. Also, we obtain the ext...
Let G be a graph with n vertices, let A(G) be an adjacency matrix of G and let PA(G,λ) be the characteristic polynomial of A(G). The adjacency spectrum of G consists of eigenvalues of A(G). A graph G is said to be determined by its adjacency s...
With geometric significance, the Euler Sombor index of a graph Γ is defined as EP(Γ)=∑{uv}∈E(Γ)d(u)2+d(v)2+d(u)d(v). It originates from the mathematical distance property and has been proven to have good chemical applicatio...
A vertex w of a connected graph G strongly resolves two distinct vertices u,v∈V(G), if there is a shortest u,w path containing v, or a shortest v,w path containing u. A set S of vertices of G is a strong resolving set for G if every two distinct...
If G is a graph, its Laplacian is the difference between the diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs G 1 and G 2 is a graph G = G 1 ⊙ u v G 2 with V ( G...
A hypercube is a graph whose nodes can be labeled by binary vectors such that the distance between the binary addresses in the graph is the Hamming distance. Due to the symmetry of the hypercube, one usually considers the graph embedded in the hyperc...
Let wG(u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, CW(G), is the number of different values of wG(u) in G, and the eccentric complexity, Cec(G), is the number of different eccentricities in G. In this pape...
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...
The definition of a Detour–Harary index is ω H ( G ) = 1 2 ∑ u , v ∈ V ( G ) 1 l ( u , v | G ) , where G is a simple and connected graph, and l ( u , v | G ) is equal to the length of the longest p...
A digraph D is an efficient open domination digraph if there exists a subset S of V ( D ) for which the open out-neighborhoods centered in the vertices of S form a partition of V ( D ) . In this work we deal with the efficient open dom...
The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline...
Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spect...
Distributed control solutions to the multi-agent systems in terms of coordination, formation, and consensus problems are extensively studied in the literature. The circular formation control of multi-agent systems becomes one of the most challenging...
The authors wish to make the following corrections to their paper [...]