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.

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

Let 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

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.

Forgotten Coindex for the Derived Sum Graphs under Cartesian Product

A topological index (TI) is a molecular descriptor that is applied on a chemical structure to compute the associated numerical value which measures volume, density, boiling point, melting point, surface tension, or solubility of this structure. It is an efficient mathematical method in avoiding laboratory experiments and time-consuming. The forgotten coindex of a structure or (molecular) graph H is defined as the sum of the degrees of all the possible pairs of nonadjacent vertices in H . For D ∈ S , R , Q , T and the connected graph H , the derived graphs D H are obtained by applying the operations S (subdivided), R (triangle parallel), Q (line superposition), and T (total graph), respectively. Moreover, a derived sum graph ( D -sum graph) is obtained by the Cartesian product of the graph H 2 with the graph D H 1 . In this study, we compute forgotten coindex of the D -sum graphs H 1 + S H 2 ( S -sum), H 1 + R H 2 ( R -sum), H 1 + Q H 2 ( Q -sum), and H 1 + T H 2 ( T -sum) in the form of various indices and coindices of the factor graphs H 1 and H 2 . At the end, we have analyzed our results using numerical tables and graphical behaviour for some particular D -sum graphs.

Omega Index of Line and Total Graphs

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.

Super H -Antimagic Total Covering for Generalized Antiprism and Toroidal Octagonal Map

Let G be a graph and H ⊆ G be subgraph of G . The graph G is said to be a , d - H antimagic total graph if there exists a bijective function f : V H ∪ E H ⟶ 1,2,3 , … , V H + E H such that, for all subgraphs isomorphic to H , the total H weights W H = W H = ∑ x ∈ V H f x + ∑ y ∈ E H f y forms an arithmetic sequence a , a + d , a + 2 d , … , a + n − 1 d , where a and d are positive integers and n is the number of subgraphs isomorphic to H . An a , d - H antimagic total labeling f is said to be super if the vertex labels are from the set 1,2 , … , | V G . In this paper, we discuss super a , d - C 3 -antimagic total labeling for generalized antiprism and a super a , d - C 8 -antimagic total labeling for toroidal octagonal map.

Analytical modeling on the coloring of certain graphs for applications of air traffic and air scheduling management

Purpose The purpose of this paper is to assess the application of graph coloring and domination to solve the airline-scheduling problem. Graph coloring and domination in graphs have plenty of applications in computer, communication, biological, social, air traffic flow network and airline scheduling. Design/methodology/approach The process of merging the concept of graph node coloring and domination is called the dominator coloring or the χ_d coloring of a graph, which is defined as a proper coloring of nodes in which each node of the graph dominates all nodes of at least one-color class. Findings The smallest number of colors used in dominator coloring of a graph is called the dominator coloring number of the graph. The dominator coloring of line graph, central graph, middle graph and total graph of some generalized Petersen graph P_(n ,1) is obtained and the relation between them is established. Originality/value The dominator coloring number of certain graph is obtained and the association between the dominator coloring number and domination number of it is established in this paper.