Total Graph Research Articles

Rainbow Mean Index of Some Classes of Graphs

In this article, we determine the rainbow mean index of cartesianproductG◻Pn, where G is a regular graph;Pm◻Pn; Wn◻K2;total graph of paths; chain of cycles and complete graphs; triangular treeand join graphs tCs∨K1.

Some results on the total zero-divisor graph of a commutative ring

PurposeThe purpose of this paper is to characterize a commutative ring R with identity which is not an integral domain such that ZT(R), the total zero-divisor graph of R is connected and to determine the diameter and radius of ZT(R) whenever ZT(R) is connected. Also, the purpose is to generalize some of the known results proved by Duric et al. on the total zero-divisor graph of R.Design/methodology/approachWe use the methods from commutative ring theory on primary decomposition and strong primary decomposition of ideals in commutative rings. The structure of ideals, the primary ideals, the prime ideals, the set of zero-divisors of the finite direct product of commutative rings is used in this paper. The notion of maximal Nagata prime of the zero-ideal of a commutative ring is also used in our discussion.FindingsFor a commutative ring R with identity, ZT(R) is the intersection of the zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R. The zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R are well studied. Hence, we determine necessary and sufficient condition so that ZT(R) agrees with the zero-divisor graph of R (respectively, agrees with the total graph induced by the set of non-zero zero-divisors of R). If Z(R) is an ideal of R, then it is noted that ZT(R) agrees with the zero-divisor graph of R. Hence, we focus on rings R such that Z(R) is not an ideal of R. We are able to characterize R such that ZT(R) is connected under the assumptions that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. With the above assumptions, we are able to determine the domination number of ZT(R).Research limitations/implicationsDuric et al. characterized Artinian rings R such that ZT(R) is connected. In this paper, we extend their result to rings R such that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. As an Artinian ring is isomorphic to the direct product of a finite number of Artinian local rings, we try to characterize R such that ZT(R) is connected under the assumption that R is ta finite direct product of rings R1, R2, … Rn with Z(Ri) is an ideal of Ri for each i between 1 to n. Their result on domination number of ZT(R) is also generalized in this paper. We provide several examples to illustrate our results proved.Practical implicationsThe implication of this paper is that the existing result of Duric et al. is applicable to large class of commutative rings thereby yielding more examples. Moreover, the results proved in this paper make us to understand the structure of commutative rings better. It also helps us to learn the interplay between the ring-theoretic properties and the graph-theoretic properties of the graph associated with it.Originality/valueThe results proved in this paper are original and they provide more insight into the structure of total zero-divisor graph of a commutative ring. This paper provides several examples. Not much work done in the area of total zero-divisor graph of a commutative ring. This paper is a contribution to the area of graphs and rings and may inspire other researchers to study the total zero-divisor graph in further detail.

Extended total graph associated to finite commutative rings

UDC 512.5 For a commutative ring R with nonzero identity 1 ≠ 0 , let Z ( R ) denote the set of zero divisors. The total graph of R denoted by T Γ ( R ) is a simple graph in which all elements of R are vertices and any two distinct vertices x and y are adjacent if and only if x + y ∈ Z ( R ) . In this paper, we define an extension of the total graph denoted by T ( Γ e ( R ) ) with vertex set as Z ( R ) , and two distinct vertices x and y are adjacent if and only if x + y ∈ Z * ( R ) , where Z * ( R ) is the set of nonzero zero divisors of R . Our main aim is to characterize the finite commutative rings whose T ( Γ e ( R ) ) has clique numbers 1,2 , and 3 . In addition, we characterize finite commutative nonlocal rings R for which the corresponding graph T ( Γ e ( R ) ) has the clique number 4.

On the topological indices and graph operation

As we know that some chemically interesting graphs can be obtained by different graph operations on some general or particular graphs, it is important to study such graph operations in order to understand how it is related to the corresponding topological indices of the original graphs. In [A. Miličević, A. Nikolić and S. Trinajstić, On reformulated Zagreb indices, Mol. Divers. 8 (2004) 393–399], authors defined two new variants of Corona product and investigated some topological indices. In this paper, we extended the work and found the formulas of the F-index, hyper-Zagreb index, and Sigma index for double join of graphs based on the total graph and double corona of graphs based on the total graph. Moreover, some applications of the derived results are also discussed.

Distance Spectra of Some Double Join Operations of Graphs

In literature, several types of join operations of two graphs based on subdivision graph, Q-graph, R-graph, and total graph have been introduced, and their spectral properties have been studied. In this paper, we introduce a new double join operation based on H1,H2-merged subdivision graph. We compute the spectrum of a special block matrix and then use it to describe the distance spectra of some double join operations of graphs. At last, we give several families of distance equienergetic graphs of diameter 3.

On Some Bounds of Degree Based Topological Indices for Total Graphs

In this paper, we discuss the concept of total graph and computed some topological indices. If $\Theta$ is a simple graph, then the elements of $\Theta$ are the vertices $\Theta_V$ and edges $\Theta_E$. For $ e=u\acute{u}\in \Theta_E$, the vertex $u$ and edge $e$, as well as $\acute{u}$ and $e$, are incident. We define the general harmonic $(GH)$ index and general sum connectivity $(GS)$ index for graph $\Theta$ regarding incident vertex-edge degrees as: $H_s^{\alpha}(\Theta)=\sum_{e\acute{u}}\big(\frac{2}{\aleph_{\acute{u}}+\aleph_{e}}\big)^{\alpha}$ and $\hat{\chi}_s^{\alpha}(\Theta)=\sum_{e\acute{u}}(\aleph_{\acute{u}}+\aleph_{e})^{\alpha}$, where $\alpha$ is any real number. In this article, we derive the closed formulas for a few standard graphs for $(GH)$ and $(GS)$ indices and then go on to calculate the lowest and the greatest general harmonic index, as well as the general sum-connectivity index, for various graphs that correspond to their total graphs.

Total graph of a lattice

Total graph of a lattice

Even vertex odd mean labeling of uniform theta graphs

Let $G$ be a graph with $p$ vertices and $q$ edges. A total graph labeling $ f:V(G)\bigcup E(G)\rightarrow \{0,1,2,3,...,2q\}$ is called even vertex odd mean labeling of a graph $G$ if the vertices of the graph $G$ label by distinct even integers from the set $\{0,2,...,2q\}$ and the labels of the edges are defined as the mean of the labels of its end vertices and these labels are $2q-1$ distinct odd integers from the set $\{1,3,5,...,2q-1\}$. In this paper we investigate the even vertex odd mean labeling of uniform theta graphs.

Gain total graphs and their spectra via G-phases and group representations

We introduce a definition for the total graph of a gain graph (Γ,ψ) on a group G by using G-phases, which are a generalization of the notion of orientation to gain graphs. Our construction is well-defined in the sense that gain graphs that are switching isomorphic have switching isomorphic total graphs. In particular, the switching equivalence class of the total graph does not depend on the particular choice of the G-phase associated with ψ. More precisely, we consider the left-right multiplication action of G on the space of all G-phases, proving that its orbits consist of sets of G-phases inducing the same switching equivalence class of gain functions on Γ or, equivalently, on its total graphs. Moreover, we prove that the stabilizers of this action are isomorphic to the centralizers of the sets of gains of closed walks in the associated gain graphs. Our construction is consistent with the existing notions of total graph for signed graphs and, in analogy with the signed case, we are able to explicitly compute the spectrum of the total graph of a regular gain graph over an arbitrary group G, where the spectrum is defined by means of a unitary representation of the gain group G.

Weak Total Coloring Conjecture and Hadwiger’s Conjecture on Total Graphs

The total graph of a graph $G$, denoted by $T(G)$, is defined on the vertex set $V(G)\cup E(G)$ with $c_1,c_2 \in V(G)\cup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to (or incident on) each other in $G$. The total chromatic number $\chi''(G)$ of a graph $G$ is defined to be the chromatic number of its total graph. The well-known Total Coloring Conjecture or TCC states that for every simple finite graph $G$ having maximum degree $\Delta(G)$, $\chi''(G)\leq \Delta(G) + 2$. In this paper, we consider two ways to weaken TCC:
 (1) Weak TCC: This conjecture states that for a simple finite graph $G$, $\chi''(G) = \chi(T(G)) \leq\Delta(G) + 3$. While weak TCC is known to be true for 4-colorable graphs, it has remained open for 5-colorable graphs. In this paper, we settle this long pending case.
 (2) Hadwiger's Conjecture for total graphs: We can restate TCC as a conjecture that proposes the existence of a strong $\chi$-bounding function for the class of total graphs in the following way: If $H$ is the total graph of a simple finite graph, then $\chi(H) \leq\omega(H) + 1$, where $\omega(H)$ is the clique number of $H$. A natural way to relax this question is to replace $\omega(H)$ by the Hadwiger number $\eta(H)$, the number of vertices in the largest clique minor of $H$. This leads to the Hadwiger's Conjecture (HC) for total graphs: if $H$ is a total graph then $\chi(H) \leq \eta(H)$. We prove that this is true if $H$ is the total graph of a graph with sufficiently large connectivity.
 A second motivation for studying Hadwiger's conjecture for total graphs is the following: Consider the class of split graphs whose vertex set is partitioned into an independent set $A$ and a clique $B$, with the following additional constraints: (1) Each vertex in $B$ has exactly 2 neighbours in $A$; (2) No two vertices in $B$ have the same neighbourhood in $A$. It is known that (European Journal of Combinatorics, 76, 159-174,2019) if Hadwiger's conjecture is proved for the squares of this special class of split graphs, then it holds also for the general case. Of course, proving the conjecture for this specialzed-looking case is indeed difficult since it is only a reformulation of the general case, and therefore it is natural to consider the difficulty level of Hadwiger's conjecture for the squares of graph classes defined by slighly modifying the above class of graphs. A natural structural modification is to assume that both $A$ and $B$ are independent sets, keeping everything else same. It turns out that the squares of this modified class of graphs is exactly the class of total graphs. From this perspective, it is not really surprising that HC on Total Graphs is also challenging. On the other hand, we show that weak TCC implies HC on total graphs. This perhaps suggests that the latter is an easier problem than the former.

Totally antimagic total labeling of ladders, prisms and generalised Petersen graphs

A total labeling is called edge-antimagic total (vertex-antimagic total) if all edge-weights (vertex-weights) are pairwise distinct. If a labeling is simultaneously edge-antimagic total and vertex-antimagic total, it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we prove that ladders, prisms and generalised Pertersen graphs are totally antimagic total graphs. We also show that the chain graph of totally antimagic total graphs is a totally antimagic total graphs.

Rainbow Vertex Connection Numbers and Total Rainbow Connection Numbers of Middle and Total Graphs

A vertex-colouring of a graph Γ is rainbow vertex connected if every pair of vertices ( u , v ) in Γ there is a u − v path whose internal vertices have different colours. The rainbow vertex connection number of a graph Γ , is the minimum number of colours needed to make Γ rainbow vertex connected, denoted by r v c ( Γ ) . Here, we study the rainbow vertex connection numbers of middle and total graphs. A total-colouring of a graph Γ is total rainbow connected if every pair of vertices ( u , v ) in Γ there is a u − v path whose edges and internal vertices have different colours. The total rainbow connection number of Γ , is the minimum number of colours required to colour the edges and vertices of Γ in order to make Γ total rainbow connected, denoted by t r c ( Γ ) . In this paper, we also research the total rainbow connection numbers of middle and total graphs.

A STUDY ON GRAPHS DEFINED ON L-SLICES

Let L be a locale with top element 1L and J be a join semilattice with bottom element 0J. The L-slice (σ,J) is the action of the locale on join semilattice satisfying certain properties. The concept of L-slices were modelled in tune with the modules in algebra. The benefit of studying L-slices is that we can approach the structure algebraically as well as topologically. This paper deals with the graph theoretic approach to L-slices. The idea of relating graphs with algebraic structures was started by the work of Beck in [3]. The algebraic properties of L-slices prompted us to consider the possibility of various graphs that could be associated with L-slices. The article introduces two different graphs on L-slices. The total graph Γ((T(σ,J)) is defined. We derive a characterisation for such graphs to be nonempty. The structural properties of Γ((T(σ,J)) is studied. The weak Zariski Topology on (σ,J) gives us the graph GT(ω∗). The conditions under which the graph is nonempty is examined. Also some of the structural properties of GT(ω∗) is obtained.

Enumeration of spanning trees in prisms of some graphs

In graph theory, a prism over a graph G is the cartesian product of the graph G with P₂. The purpose of this work is to investigate the complexity of the prisms of some path and cycle-related graphs. In particular, we obtain simpler and more explicit formulas for the complexity of a special class of prisms of path-related graphs: fan graph, ladder graph, the composition Pn[P₂] graph, and book graph. Moreover, we obtain straightforward formulas for the complexity of a special class of prisms of cycle-related graphs: wheel graph, gear graph, prism graph, n−crossed prism graph, mirror graph M(Cn) of even cycle Cn, twisted prism, total graph T(Cn) of the cycle Cn, the friendship graph, the flower graph, and planner sunflower graph. These closed formulas are deduced using some basic properties of block matrix, recurrence relation, eigenvalues of circulant matrices, and orthogonal polynomials.

The Forcing Metric Dimension of a Total Graph of Nonzero Annihilating Ideals

The Forcing Metric Dimension of a Total Graph of Nonzero Annihilating Ideals

A vulnerability detection algorithm based on residual graph attention networks for source code imbalance (RGAN)

It is essential to detect potential vulnerabilities in software to ensure its safety. As software systems become more complex, traditional static vulnerability detection methods perform poorly. Currently, deep learning-based vulnerability detection models only extract source code vulnerability features using sequences or graphs. Sequential neural networks ignore structural information in the code, such as control flow diagrams and data flow diagrams. Additionally, graph neural networks cannot accurately extract features due to the lack of effective methods for extracting nodes’ features and aggregating global information. To address the above issue, we propose a vulnerability detection algorithm based on residual graph attention networks for source code imbalance (RGAN). Firstly, a local feature extraction module (PE-BL-A module) is designed. Using the sequence neural network, the module extracts various useful features, including node features in a control flow diagram based on local semantic features. Secondly, we present the Residual Graph Attention Network module (RGAT). To learn and update node features along the control flow direction, the module uses a graph attention network with residual connections. In this module, a mean biaffine attention pooling mechanism is proposed that can extract total graph vulnerability features more effectively. Thirdly, a dynamic cross-entropy loss function is designed. Using this function, it can handle sample imbalances during training. Finally, experiments conducted on several benchmark datasets demonstrate that the proposed model achieves state-of-the-art results.

The most general structure of graphs with hamiltonian or hamiltonian connected square

On the basis of recent results on hamiltonicity, [5], and hamiltonian connectedness, [9], in the square of a 2-block, we determine the most general block-cutvertex structure a graph G may have in order to guarantee that G2 is hamiltonian, hamiltonian connected, respectively. Such an approach was already developed in [10] for hamiltonian total graphs.

Erratum. Dominator coloring of total graph of path and cycles

Erratum. Dominator coloring of total graph of path and cycles

Note for the conjecture on the generalized 4-connectivity of total graphs of the complete bipartite graph

The generalized k-connectivity κk(G) of a graph G, introduced by Hager in 1985, is a natural generalization of the concept of connectivity κ(G), which is just for k=2. Rongxia Hao et al. determined the generalized 4-connectivity of the total graphs of the complete equipartition bipartite graph and conjectured that κ4(T(Km,m+1))=2m−1 and κ4(T(Km,n))=2m for n>m+1 and m≥4 in [Appl. Math. Comput. 422 (2022)]. In this paper, we solved this conjecture.

Dominator Coloring of Total Graph of Path and Cycle

A dominator coloring of a graph G is a proper coloring in which every vertex of G dominates every vertex of at least one-color class possibly its own class and each color class is dominated by at least one vertex. The minimum number of colors required for dominator coloring of G is called the dominator chromatic number of G and is denoted by χdG. In this paper, we have established the relation between dominator chromatic number χdG, chromatic number χG and domination number γG. We have investigated results on total graphs of path and cycle with χdG = χ G + γ G and χdG = χ G + γ G - 1.