Total Graph Research Articles

The total irregularity strength of m copies of the friendship graph

This paper deals with the totally irregular total labeling of the disjoin union of friendship graphs. The results shows that the disjoin union of copies of the friendship graph is a totally irregular total graph with the exact values of the total irregularity strength equals to its edge irregular total strength.

The Reformulated F-Index of Vertex and Edge F-Join of Graphs

The valency based topological indices (TIs) are defined by algebraic functions as the chemical graph theory tools in which the structural parameters are used as input and the output is related with the topology of chemical species. In theoretical chemistry, the TIs are mainly used to investigate/develop the QSAR and QSPR investigations of the molecular graphs. The Reformulated F-index or RFI is one such kind of TIs. The RFI for a molecular graph Y is defined as R F Y = ∑ u v ∈ E Y d Y u + d Y v − 2 3 , where d Y u is the degree of a vertex u in Y . In this paper, we study the RFI for vertex and edge version of F-join of given general graphs which are related to subdivision and total graph.

On the construction of super edge-magic total graphs

Suppose G = ( V , E ) be a simple graph with p vertices and q edges. An edge-magic total labeling of G is a bijection f : V ∪ E → {1, 2, …, p + q } where there exists a constant r for every edge x y in G such that f ( x )+ f ( y )+ f ( x y )= r . An edge-magic total labeling f is called a super edge-magic total labeling if for every vertex v ∈ V ( G ) , f ( v )≤ p . The super edge-magic total graph is a graph which admits a super edge-magic total labeling. In this paper, we consider some families of super edge-magic total graph G . We construct several graphs from G by adding some vertices and edges such that the new graphs are also super edge-magic total graphs.

Some Cohen–Macaulay graphs arising from finite commutative rings

For a given finite commutative ring [Formula: see text] with [Formula: see text], one may associate a graph which is called the total graph of [Formula: see text]. This graph has [Formula: see text] as the vertex set and its two distinct vertices [Formula: see text] and [Formula: see text] are adjacent exactly whenever [Formula: see text] is a zero-divisor of [Formula: see text]. In this paper, we give necessary and sufficient conditions for two classes of total graphs to be Cohen–Macaulay.

The IRC Indices of Transformation and Derived Graphs

An irregularity index IR(Γ) of a graph Γ is a nonnegative numeric quantity (i.e., IR(Γ)≥0) such that IR(Γ)=0 iff Γ is a regular graph. In this paper, we show that IRC closely correlates with the normal boiling point Tbp and the standard heat of formation ΔHfo of lower benzenoid hydrocarbons. The correlation models that fit the data efficiently for both Tbp and ΔHfo are linear. We develop further mathematical properties of IRC by calculating its exact expressions for the recently introduced transformation graphs as well as certain derived graphs, such as the total graph, semi-total point graph, subdivision graph, semi-total line graph, double, strong double, and extended double cover graphs. Some open problems are proposed for further research on the IRC index of graphs.

HZ-Coindex for the Sum-Graphs under Cartesian Product

:Let G = (V (G), E(G)) be a simple connected graph with V (G) and E(G) as the vertex and edge sets respectively. The physical and structural properties of the graphs are discussed with different mathematical tools, One of them is topological index (TI) and it can be obtained directly from molecular structures and rapidly computed for large number of molecules. Graph operations play an important role in the development of various new classes of graphs. If G is a connected simple graph then we obtained new graphs by applying various operations on G such as S subdivided, R triangle parallel, Q line superposition and T total graphs operations on G. In this paper, we computed the hyper Zagreb coindex of derived graphs in the form of their factor graphs. At the end, we have analyzed our results by the numerical tables for the particular generalized D-sum graphs

Pairs of maps preserving singularity on subsets of matrix algebras

Let F be an algebraically closed field and Mn be the n×n matrix algebra over F. A total graph of the full matrix algebra is the graph with Mn as vertices, and two distinct matrices A,B are adjacent if and only if A+B is singular. The characterization of all the automorphisms of the total graph is an open question. Motivated by this problem, we study pairs of maps on a subset of Mn preserving the singularity of matrix pencils A+λB. In particular, we characterize maps T1,T2:Mn→Mn satisfying the condition A+λB is singular if and only if T1(A)+λT2(B) is singular, for any A,B∈Mn and any non-zero λ∈F. Namely, we prove that in this case T1=T2 and they are of the form T1(A)=T2(A)=PAQ for all A∈Mn, or of the form T1(A)=T2(A)=PAtQ for all A∈Mn, where P,Q∈Mn are non-singular matrices.

Internally disjoint trees in the line graph and total graph of the complete bipartite graph

Let G be a connected graph, S⊆V(G) and |S|≥2, a tree T in G is called an S-tree if S⊆V(T). Two S-trees T1 and T2 are called internally disjoint if E(T1)∩E(T2)=∅ and V(T1)∩V(T2)=S. For an integer r with 2≤r≤n, the generalizedr-connectivityκr(G) of a graph G is defined as κr(G)=min{κG(S)|S⊆V(G) and |S|=r}, where κG(S) denotes the maximum number k of internally disjoint S-trees in G. In this paper, we consider the generalized 4-connectivity of the line graph L(Km,n) and total graph T(Km,n) of the complete bipartite graph Km,n with 2≤m≤n. The results that κ4(L(Km,n))=m+n−3 for 2≤m≤3 and κ4(L(Km,n))=m+n−4 for m≥4 are obtained by determining κ4(Km×Kn). In addition, we obtain that κ4(T(Km,m))=δ(T(Km,m))−2=2m−2 for m≥2. These results improve the known results about the generalized 3-connectivity of L(Km,n) and T(Km,m) in [Appl. Math. Comput. 347 (2019) 645–652].

First General Zagreb Co‐Index of Graphs under Operations

Topological indices are graph‐theoretic parameters which are widely used in the subject of chemistry and computer science to predict the various chemical and structural properties of the graphs respectively. Let G be a graph; then, by performing subdivision‐related operations S, Q, R, and T on G, the four new graphs S(G) (subdivision graph), Q(G) (edge‐semitotal), R(G) (vertex‐semitotal), and T(G) (total graph) are obtained, respectively. Furthermore, for two simple connected graphs G and H, we define F‐sum graphs (denoted by G+FH) which are obtained by Cartesian product of F(G) and H, where F ∈ {S, R, Q, T}. In this study, we determine first general Zagreb co‐index of graphs under operations in the form of Zagreb indices and co‐indices of their basic graphs.

ON SOME TOTAL GRAPHS ON FINITE RINGS

We give a decomposition of total graphs on some finite commutative rings R = Zm, where the set of zero-divisors of R is not an ideal. In particular, we study the total graph T(􀀀(Z2npm))where p is a prime and m and n are positive integers and investigate some graph theoretical properties with some of its fundamental subgraphs.

On the total domination number of total graphs

On the total domination number of total graphs

New classes of graphs with edge $ \; \delta- $ graceful labeling

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

Rationale for the Choice of the Ellipsoidal Reflector Parameters for Biomedical Photometers

Biomedical photometersʼ information-measuring systems with ellipsoidal reflectors have acceptable results in determining of biological tissues optical properties in the visible and near-infrared spectral range. These photometers make it possible to study the optical radiation propagation in turbid media for direct and inverse problems of light-scattering optics. The purpose of this work is to study the influence of the ellipsoidal reflectors design parameters on the results of biomedical photometry when simulating the optical radiation propagation in a system of biological tissue and reflectors in transmitted and reflected light.The paper substantiates the choice of the ellipsoidal reflectors’ focal parameter for efficient registration of forward and backscattered light. The methodology of the process is illustrated by the results of a model experiment using the Monte Carlo simulation for samples of human brain white and gray matter at the visible range of 405 nm, 532 nm, and 650 nm. The total transmittance, diffuse reflectance, and absorption graphs depending on the sample thickness were obtained. Based on the introduced concepts of the ellipsoidal reflector efficiency index and its efficiency factor, the expediency of choosing the ellipsoidal reflectors focal parameter is analyzed to ensure the registration of the maximum amount of scattered light. The graphs of efficiency index in reflected and transmitted light for different thickness samples of white and gray matter and efficiency factors depending on the sample thickness were obtained.The influence of the reflectors ellipticity on the illuminance of various zones of photometric images using the example of an absorbing biological medium – pig liver tissue – at wavelength of 405 nm with a Monte Carlo simulation was analyzed.The optical properties of biological media (scattering and absorption coefficients, scattering anisotropy factor, refractive index) and the samples’ geometric dimensions, particularly the thickness, are predetermined when choosing the ellipsoidal reflectors parameters for registration of the scattered light. Coordinates of the output of photons and their statistical weight obtained in the Monte Carlo simulation of light propagation in biological tissue have a physical effect on a characteristic scattering spot formation in the receiving plane of a biomedical photometer with ellipsoidal reflectors.

An Ideal-Based Dot Total Graph of a Commutative Ring

In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail.

On Topological Indices of Total Graph and Its Line Graph for Kragujevac Tree Networks

Kragujevac tree is indicated by K ; K ∈ K g q = s 2 t + 1 + 1 , s with order and size s 2 t + 1 + 1 and s 2 t + 1 , respectively. In this paper, we have a look at certain topological features of the total graph and line graph of the total graph of the considered tree, i.e ., Kragujevac tree, by computing different topological indices and polynomials.

An ideal-based graph of a bounded lattice

Let [Formula: see text] be a lattice, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] the set of all elements of [Formula: see text] which are not prime to [Formula: see text]. In this paper, we investigate some interesting properties of [Formula: see text] and introduce the total graph of a lattice [Formula: see text] with respect to an ideal [Formula: see text]. It is the graph with all elements of [Formula: see text] as vertices and for distinct [Formula: see text], the vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. We investigated the basic properties of this graph and its induced subgraphs as diameter, girth, clique number, cut vertex and independence number.

ON THE STRONG METRIC DIMENSION OF A TOTAL GRAPH OF NONZERO ANNIHILATING IDEALS

AbstractLet R be a commutative ring with identity which is not an integral domain. An ideal I of R is called an annihilating ideal if there exists $r\in R- \{0\}$ such that $Ir=(0)$ . The total graph of nonzero annihilating ideals of R is the graph $\Omega (R)$ whose vertices are the nonzero annihilating ideals of R and two distinct vertices $I,J$ are joined if and only if $I+J$ is also an annihilating ideal of R. We study the strong metric dimension of $\Omega (R)$ and evaluate it in several cases.

On the r-dynamic coloring of some fan graph families

Abstract In this paper, we determine the r-dynamic chromatic number of the fan graph Fm,n and determine sharp bounds of this graph invariant for four related families of graphs: The middle graph M(Fm,n ), the total graph T (Fm,n ), the central graph C(Fm,n ) and the line graph L(Fm,n ). In addition, we determine the r-dynamic chromatic number of each one of these four families of graphs in case of being m = 1.

Properties of Total Transformation Graphs for General Sum-Connectivity Index

The study of networks and graphs through structural properties is a massive area of research with developing significance. One of the methods used in studying structural properties is obtaining quantitative measures that encode structural data of the whole network by the real number. A large collection of numerical descriptors and associated graphs have been used to examine the whole structure of networks. In these analyses, degree-related topological indices have a significant position in theoretical chemistry and nanotechnology. Thus, the computation of degree-related indices is one of the successful topics of research. The general sum-connectivity GSC index of graph Q is described as χ ρ Q = ∑ qq ′ ∈ E Q d q + d q ′ ρ , where d q presents the degree of the vertex q in Q and ρ is a real number. The total graph T Q is a graph whose vertex set is V Q ∪ E Q , and two vertices are linked in T Q if and only if they are either adjacent or incident in Q . In this article, we study the general sum-connectivity index χ ρ Q of total graphs for different values of ρ by using Jensen’s inequality.

Allocation of steady-state areas of total electrical load schedules of 10/0.4 kv transformer substations located in residential areas

Improving the efficiency of regulating the parameters of distribution network modes is an actual task for the energy sector, requiring a detailed analysis of the nature of the total electrical load graphs of 10/0.4 kV transformer substations. The complexity of the problem in consideration is due to the fact that the load variation of transformer substations 10/0.4 kV in residential areas occurs randomly, due to the significant number, wide range and probabilistic nature of the operating modes of connected consumers, and the lack of deterministic links between consumers of electricity. The investigated function of total load of transformer substations 10/0.4 kV on a daily interval is non-stationary, in this connection there was a necessity of allocation of stationarity areas for adjusting the process of regulation of parameters of electrical network modes. In order to solve this problem, total load graphs for 10/0.4 kV transformer substations located in a residential area were constructed based on experimental measurements of active and reactive loads. The use of these dependencies made it possible to analyse the distribution law for the active and reactive power of the total load of 10/0.4 kV transformer substations, which change in time is stochastic characterised. The use of probabilistic-statistical modelling was justified to reliably describe the load variation process of 10/0.4 kV transformer substations. The hypothesis of a normal law distribution of the functions under consideration was confirmed and parametric tests were performed. Fisher's F-criterion was used to confirm the hypothesis of variance constancy, and Student's t-criterion was used to confirm the hypothesis of mathematical expectation constancy. Using the fact of constancy of dispersions and expectations of investigated mode parameters, autocorrelation coefficients of investigated random functions were determined and autocorrelation function graphs were plotted. In order to approximate the functions under investigation, the autocorrelation function coefficients were determined using the least-squares method, and an analysis of the attenuation of the autocorrelation function was carried out. The calculations carried out have highlighted the areas of stationarity on the total load curves of the 10/0.4 kV transformer substations. These stationarity plots can be used for further modelling of load graphs and the formation of control actions to adjust the load of consumer regulators in order to equalise the overall electrical load graph, as well as for voltage regulation facilities, which will contribute to the required modes of operation of the electricity distribution networks.