Sum Graph Research Articles

On regular sets in Cayley sum graphs

On regular sets in Cayley sum graphs

On rings whose prime ideal sum graphs are line graphs

Let [Formula: see text] be a commutative ring with unity. The prime ideal sum graph of the ring [Formula: see text] is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph.

Whole Even Sum Domination in Graphs

In this paper, a new concept of domination which is called whole even sum domination is presented, many bounded, properties, remarks, examples, are introduced. Furthermore, some operations as join and corona graphs of sum graphs are discussed. Finally, determined this number of the graph obtained from a graph G by the removal of all edges between any pair of neighbours of a vertex.

Strong Metric Dimension of the Prime Ideal Sum Graphs of Commutative Rings

Strong Metric Dimension of the Prime Ideal Sum Graphs of Commutative Rings

Bounds For Spectral Radius and Energy of $PIS$ Graphs

Once the spectral radius and energy of a graph structure have been defined, many properties have been studied. The spectral radius and energy of a graph are related to the eigenvalues of the adjacency matrix of the graph. In this paper, we define an adjacency matrix for a prime ideal sum ($PIS$) graph and then extend the concepts of spectral radius and energy to $PIS$ graphs. Some bound theorems on the energy and spectral radius of $PIS$ graph structures are given. A SageMath code for plotting these graphs is also provided.

Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs

Let R be a commutative ring with unity. The prime ideal sum graph PIS ( R ) of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent if and only if I + J is a prime ideal of R. In this paper, we study some interplay between algebraic properties of rings and graph-theoretic properties of their prime ideal sum graphs. In this connection, we classify non-local commutative Artinian rings R such that PIS ( R ) is of crosscap at most two. We prove that there does not exist a non-local commutative Artinian ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative Artinian rings of genus one prime ideal sum graphs.

Subspace Sum Graphs over Residue Class Rings

Subspace Sum Graphs over Residue Class Rings

Code and Data Repository for Solving the Minimum Sum Coloring Problem: Alternative Models, Exact Solvers, and Metaheuristics

This repository provides data for the problem and scripts for the models. The main folders are 'data', 'scripts', and 'results'. 'data': This folder includes minimum sum graph coloring data sets. 'src': Our metaheuristic solver Q-Card can be found in this folder. 'scripts': This folder provides a running script. 'results': This folder provides the results.

Every Tree is an Integral Sum Graph

A finite simple graph G is called an integral sum graph (respectively, sum graph) if there is a bijection f from the vertices of G to a set of integers S (respectively, a set of positive integers S) such that uv is an edge of G if and only if f (u)+f (v) ∈ S. In 1999, Liaw et al (Ars Comb., Vol.54, 259-268) posed the conjecture that every tree is an integral sum graph. In this note, we prove that all trees are integral sum graphs. Further, we prove that every bipartite graph is an induced subgraph of a sum graph G with sum number σ(G) = 1.

Forgotten Topological and Wiener Indices of Prime Ideal Sum Graph of Zn.

Chemical graph theory is a sub-branch of mathematical chemistry, assuming each atom of a molecule is a vertex and each bond between atoms as an edge. Owing to this theory, it is possible to avoid the difficulties of chemical analysis because many of the chemical properties of molecules can be determined and analyzed via topological indices. Due to these parameters, it is possible to determine the physicochemical properties, biological activities, environmental behaviours and spectral properties of molecules. Nowadays, studies on the zero divisor graph of Z_n via topological indices is a trending field in spectral graph theory. For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R and two distinctvertices I and J are adjacent if and only if I+J is a prime ideal of R. In this study the forgotten topological index and Wiener index of the prime ideal sum graph of Z_n are calculated for n=p^α,pq,p^2 q,p^2 q^2,pqr,p^3 q,p^2 qr,pqrs where p,q,r and s are distinct primes and a Sage math code is developed for designing graph and computing the indices. In the light of this study, it is possible to handle the other topological descriptors for computing and developing new algorithms for next studies and to study some spectrum and graph energies of certain finite rings with respect to PIS-graph easily.

Subgroup total perfect codes in Cayley sum graphs

Subgroup total perfect codes in Cayley sum graphs

Quasi-Laplacian energy of $$\psi $$-sum graphs

Quasi-Laplacian energy of $$\psi $$-sum graphs

Metric dimension in a prime ideal sum graph of a commutative ring

The prime ideal sum graph of a commutative unital ring [Formula: see text], denoted by [Formula: see text], is an undirect and simple graph whose vertices are nontrivial ideals of [Formula: see text] and there exists an edge between two distinct vertices if and only if their sum is a prime ideal of [Formula: see text]. In this paper, the metric dimension of [Formula: see text] is discussed and some formulae for this parameter in [Formula: see text] are given.

Computing Topological Descriptors of Prime Ideal Sum Graphs of Commutative Rings

Let n≥1 be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of Zn, where n=pα,p2q,p2q2,pqr,p3q,p2qr, and pqrs for the different prime integers p,q,r, and s. Moreover, we construct M-polynomials and CoM-polynomials using the PIS-graph structure of Zn to avoid the difficulty of computing the descriptors via formulas directly. Furthermore, we present a geometric comparison for representations of each surface obtained by M-polynomials and CoM-polynomials. Finally, we discuss the applicability of algebraic graphs to chemical graph theory.

Some applications of eigenvalues of unitary Cayley graphs of matrix rings over finite fields

ABSTRACT In this work, we provide two applications of the eigenvalues of the unitary Cayley graphs over matrix rings over finite fields. For a ring R, denotes the Jacobson radical of R. In 2012, Kiani and Aghaei conjectured that if R and S are finite rings and their unitary Cayley graphs are isomorphic, then and are isomorphic. If this conjecture holds and , then R is characterized by its unitary Cayley graph, and we say that R is a ring determined by unitary Cayley graphs. Kiani and Aghaei showed that every finite commutative ring and are such rings. We examine the eigenvalues of and obtain many new families of rings determined by unitary Cayley graphs. In 2021, Podestá and Videla characterized all finite commutative rings R such that the graphs in the triple are mutually equienergetic non-isospectral and Ramanujan where is the unitary Cayley sum graph. We use the eigenvalues of the graph and some observation on the number of matrices of the given rank to extend Podestá and Videla's results by working on the finite non-commutative ring being a product of matrix rings over local rings.

Fast winning strategies for Staller in the Maker–Breaker domination game

The Maker–Breaker domination game is played on a graph G by two players, called Dominator and Staller, who alternately choose a vertex that has not been played so far. Dominator wins the game if his moves form a dominating set. Staller wins if she plays all vertices from a closed neighborhood of a vertex v∈V(G). Dominator’s fast winning strategies were studied earlier. In this work, we concentrate on the cases when Staller has a winning strategy in the game. We introduce the invariant γSMB′(G) (resp., γSMB(G)) which is the smallest integer k such that, under any strategy of Dominator, Staller can win the game by playing at most k vertices, if Staller (resp., Dominator) plays first on the graph G.We prove some basic properties of γSMB(G) and γSMB′(G) and study the parameters’ changes under some operators as taking the disjoint union of graphs or deleting a cut vertex. We show that the inequality δ(G)+1≤γSMB′(G)≤γSMB(G) always holds and that for every three integers r,s,t with 2≤r≤s≤t, there exists a graph G such that δ(G)+1=r, γSMB′(G)=s, and γSMB(G)=t. We prove exact formulas for γSMB′(G) where G is a path or it is a tadpole graph which is obtained from the disjoint union of a cycle and a path by adding one edge between them.

Characterization of rings with genus two prime ideal sum graphs

Let [Formula: see text] be a commutative ring with unity. The prime ideal sum graph of the ring [Formula: see text] is a simple undirected graph whose vertex set is the set of nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we characterize all the finite non-local commutative rings whose prime ideal sum graph is of genus [Formula: see text].

Integral sum graphs Gn and G-r,n are perfect graphs

A graph G is an integral sum graph (sum graph) if its vertices can be labeled with distinct integers (positive integers) so that e = uv is an edge of G if and only if the sum of the labels on vertices u and v is also a label in G. A graph G is perfect if the chromatic number of each of its induced subgraphs is equal to the clique number of the same. A simple graph G is of class 1 if its edge chromatic number and maximum degree are same. In this paper, we prove that integral sum graphs Gn , G 0 , n and G − r , n over the label sets [ 1 , n ] , [ 0 , n ] and [ − r , n ] , respectively, are perfect graphs as well as of class 1 for r , n ∈ N . We also obtain a few structural properties of these graphs.

The Space Complexity of Sum Labelling

A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider combinatorial questions like the minimum number of isolated vertices that need to be added to a given graph to make it a sum graph. In this paper, we initiate the study of sum graphs from the viewpoint of computational complexity. Notice that every n-vertex sum graph can be represented by a sorted list of n positive integers where edge queries can be answered in mathscr {O}(log n) time. Therefore, upper-bounding the numbers used as vertex labels also upper-bounds the space complexity of storing the graph in the database. We show that every n-vertex, m-edge, d-degenerate graph can be made a sum graph by adding at most m isolated vertices to it, such that the largest numbers used as vertex labels grows as mathscr {O}(n^2d). This enables us to store the graph using mathscr {O}(mlog n) bits of memory. For sparse graphs (graphs with mathscr {O}(n) edges), this matches the trivial lower bound of Omega (nlog n). As planar graphs and forests have constant degeneracy, our result implies an upper bound of mathscr {O}(n^2) on their label numbers. The previously best known upper bound on the numbers needed for labelling general graphs with the minimum number of isolated vertices was mathscr {O}(4^n), due to Kratochvíl, Miller & Nguyen (2001). Furthermore, their proof was existential, whereas our labelling can be constructed in polynomial time.

ISI(G) Topological Index of Sum Graphs For Some Finite Groups

ISI(G) Topological Index of Sum Graphs For Some Finite Groups