An independent Roman {2}-dominating function (IR2DF) f : V -> {0,1, 2} in a graph G = (V, E) has the properties that Sigma(u is an element of N(v)) f(u) >= 2 if f(v) = 0, and f(u) = 0 for u is an element of N(v) if f(v) >= 1, where v is an element of V. The weight of an IR2DF in a graph G is defined as the sum of its function value for all vertices, given by omega(f) = Sigma(v is an element of V) f(v). The independent Roman {2}-domination number of G, denoted i{R2}(G), is the minimum weight of an IR2DF on G. In this paper, we prove that the independent Roman {2}-domination problem (IR2D) is NP-complete, even when restricted to chordal bipartite graphs. We then give an exact formula for the IR2D in corona graphs. Finally, we present two linear-time algorithms for solving IR2D for proper interval graphs and block-cactus graphs, respectively.
Let G be a simple connected graph with diameter d, and k is an element of [1, d] be an integer. A radio k-coloring of graph G is a mapping g : V(G) -> {0} boolean OR N satisfying vertical bar g(u) - g(v)vertical bar >= 1 + k d(u, v) for any pair of distinct vertices u and v of the graph G, where d(u, v) denotes distance between vertices u and v in G. The number max{g(u) : u is an element of V (G)g is known as the span of g and is denoted by rc(k)(g). The radio k-chromatic number of graph G, denoted by rc(k)(G), is defined as min{rc(k)(g) : g is a radio k-coloring of G}. For k = d - 1 the radio k-coloring of graph G is called an antipodal coloring. So rc(d-1) is called the antipodal number of G and is denoted by ac(G). Here, we study antipodal coloring of the Cartesian product of the complete graph K-r and cycle C-s, K-r square C-s, for r >= 4 and s >= 3. We determine the antipodal number of K-r square C-s, for even r >= 4 with s 1 (mod 4), and for any r >= 4 with s = 4t + 2, t odd. Also, for the remaining values of r and s, we give lower and upper bounds for ac(K-r square C-s).
A numbering f of a graph G of order n is a labeling that assigns distinct elements of the set {1, 2, ... , n} to the vertices of G. The strength str (G) of G is defined by str (G) = min {strf (G) |f is a numbering of G }, where strf (G) = max {f (u) + f (v) |uv is an element of E (G) }. Using the concept of independence number of a graph, we determine formulas for the strength of powers of paths and cycles. To achieve the latter result, we establish a sharp upper bound for the strength of a graph in terms of its order and independence number and a formula for the independence number of powers of cycles.
A subset S of vertices of a graph G is in general position if no shortest path in G contains three vertices of S. The general position problem consists of finding the number of vertices in a largest general position set of G, whilst the lower general position problem asks for a smallest maximal general position set. In this paper we determine the lower general position numbers of several families of Cartesian products. We also show that the existence of small maximal general position sets in a Cartesian product is connected to a special type of general position set in the factors, which we call a terminal set, for which adding any vertex u from outside the set creates three vertices in a line with u as an endpoint. We give a constructive proof of the existence of terminal sets for graphs with diameter at most three. We also present conjectures on the existence of terminal sets for all graphs and a lower bound on the lower general position number of a Cartesian product in terms of the lower general position numbers of its factors.
For n >= 2t + 1 where t >= 1, the circulant graph C-n(1, 2, ... , t) consists of the vertices v(0), v(1), v(2), ... , v(n-1) and the edges v(i)v(i+1), v(i)v(i+2), ... , v(i)v(i+t), where i = 0, 1, 2, ... , n - 1, and the subscripts are taken modulo n. We prove that the metric dimension dim(C-n(1, 2, ... , t)) >= inverted right perpendicular2t/3inverted left perpendicular + 1 for t >= 5, where the equality holds if and only if t = 5 and n = 13. Thus dim(C-n(1, 2, ... , t)) >= inverted right perpendicular2t/3inverted left perpendicular + 2 for t >= 6. This bound 3 is sharp for every t >= 6.
In this paper, we introduce the zero-divisor associate graph Gamma D(R) over a finite commutative ring R. It is a simple undirected graph whose vertex set consists of all non-zero elements of R, and two vertices a, b are adjacent if and only if there exist non-zero zero-divisors z1, z2 in R such that az1 = bz2. We determine the necessary and sufficient conditions for connectedness and completeness of Gamma D(R) for a unitary commutative ring R. The chromatic number of Gamma D(R) is also studied. Next, we characterize the rings R for which Gamma D(R) becomes a line graph of some graph. Finally, we give the complete list of graphs with at most 15 vertices which are realizable as Gamma D(R), characterizing the associated ring R in each case.
The folded hypercube FQ(n) is the Cayley graph Cay(Z(2)(n), S), where S = {e(1), e(2), ... , e(n)} U {u = e(1) + e(2) + center dot center dot center dot + e(n)}, and e(i) = (0, ... , 0,1, 0, ... , 0), with 1 at the ith position, 1 <= i <= n. In this paper, we show that the folded hypercube FQn is a distance-transitive graph. Then, we study some properties of this graph. In particular, we show that if n >= 4 is an even integer, then the folded hypercube FQ(n) is an automorphic graph, that is, FQ(n) is a distance-transitive primitive graph which is not a complete or a line graph.
Let R be a finite commutative ring with or without unity and Gamma e(R) be its extended zero-divisor graph with vertex set Z*(R) = Z(R) \ {0} and two distinct vertices x, y are adjacent if and only if x.y = 0 or x + y E Z*(R). In this paper, we characterize finite commutative rings whose extended zero-divisor graph have clique number 1 or 2. We completely characterize the rings of the form R similar to= R1 X R2, where R1 and R2 are local, having clique number 3, 4 or 5. Further we determine the rings of the form R similar to= R1 X R2 X R3, where R1,R2 and R3 are local rings, to have clique number equal to six.
For any graph, Weisfeiler and Leman assigned the smallest matrix algebra which contains the adjacency matrix of the graph. The coherent configuration underlying this algebra for a graph Gamma is called the coherent configuration of Gamma, denoted by X (Gamma). In this paper, we study the coherent configuration of circular-arc graphs. We give a characterization of the circular-arc graphs Gamma, where X(Gamma) is a homogeneous coherent configuration. Moreover, all homogeneous coherent configurations which are obtained in this way are characterized as a subclass of Schurian coherent configurations.
In this article, we discussed the zero-divisor graph of a commutative ring with identity Fp + uFp + u2Fp where u3 = 0 and p is an odd prime. We find the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth of a zero-divisor graph associated with the ring. We find some of topological indices and the main parameters of the code derived from the incidence matrix of the zerodivisor graph Gamma(R). Also, we find the eigenvalues, energy and spectral radius of both adjacency and Laplacian matrices of Gamma(R).
Let G be a finite group. The directed inclusion graph of cyclic subgroups of G, Ic(G), is the digraph with vertices of all cyclic subgroups of G, and for two distinct cyclic subgroups hai and hbi, there is an arc from hai to hbi if and only if hbi subset of hai. The (undirected ) inclusion graph of cyclic subgroups of G, Ic(G), is the underlying graph of -Ic ->(G), that is, the vertex set is the set of all cyclic subgroups of G and two distinct cyclic subgroups hai and hbi are adjacent if and only if hai subset of hbi or hbi subset of hai. In this paper, we first show that, if G and H are finite groups such that Ic(G) similar to= Ic(H) and G is cyclic, then H is cyclic. We show that for two cyclic groups G and H of orders p alpha 11 ... p alpha tt and q beta 11 ... q beta s and only if t = s and by a suitable sigma, alpha i = beta sigma(i). Also for any cyclic groups G, H, if Ic(G) similar to= Ic(H), then Ic(G) similar to= ->- ->- Ic(H). We also show that for two finite abelian groups G and H, Ic(G) similar to= Ic(H) if and only if |x-(G)| = |x-(H)| and by a convenient permutation the graph of their sylow subgroups are isomorphic. In this case, their directed inclusion graphs are isomorphic too.
The complement of the intersection graph of subgroups of a group G, denoted by 5c(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and its two distinct vertices H and K are adjacent if and only if H & AND; K = 1, where 1 denotes the trivial subgroup of G. In this paper, we classify all finite groups whose complement of the intersection graph of subgroups is one of totally disconnected, bipartite, complete bipartite, tree, star graph or C3-free. Also we characterize all the finite groups whose complement of the intersection graph of subgroups is planar.
A vertex u of a graph G = ( V , E ), ve -dominates every edge incident to u , as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (or a ved-set for short) if every edge of E is ve- dominated by at least one vertex of S . The vertex-edge domination number is the minimum cardinality of a ved-set in G . In this paper, we investigate the graphs having unique minimum ved-sets that we will call UVED-graphs. We start by giving some basic properties of UVED-graphs. For the class of trees, we establish two equivalent conditions characterizing UVED-trees which we subsequently complete by providing a constructive characterization.
In 2020, mathematical chemist, Ivan Gutman, introduced a new vertex-degree-based topological index called the Sombor Index, denoted by SO (G), where G is a simple, connected, finite, graph. This paper aims to present some novel formulas, along with some upper and lower bounds on the Sombor Index of generalized Sierpinski graphs; originally defined by Klavzar and Milutinovic by replacing the complete graph appearing in S (n, k) with any graph and exactly replicating the same graph, yielding self-similar graphs of fractal nature; and on the Sombor Index of the m-Mycielskian or the generalized Mycielski graph; formed from an interesting construction given by Jan Mycielski (1955); of some simple graphs such as K-n, C-n(2), C-n, and P-n. We also provide Python codes to verify the results for the SO (S (n, K-m)) and SO (mu(m) (K-n)).
Let g = (V, 6) be a simple graph, an L(2,1)-labeling of g is an assignment of labels from non-negative integers to vertices of g such that adjacent vertices get labels which differ by at least by two, and vertices which are at distance two from each other get different labels. The lambda-number of g, denoted by lambda(g), is the smallest positive integer P such that g has an L(2,1)-labeling with all labels as members of the set {0,1, ... ,P}. The zero-divisor graph of a finite commutative ring R with unity, denoted by Gamma(R), is the simple graph whose vertices are all zero divisors of R in which two vertices u and v are adjacent if and only if uv = 0 in R. In this paper, we investigate L(2,1)-labeling of some zero-divisor graphs. We study the partite truncation, a graph operation that allows us to obtain a reduced graph of relatively small order from a graph of significantly larger order. We establish the relation between lambda-numbers of the graph and its partite truncated one. We make use of the operation partite truncation to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring.
The commuting graph of a finite non-commutative semigroup S, denoted by Δ(S), is the simple graph whose vertices are the non-central elements of S and two distinct vertices x; y are adjacent if xy = yx. In the present paper, we study various graph-theoretic properties of the commuting graph Δ(B_n) of Brandt semigroup B_n including its diameter, clique number, chromatic number, independence number, strong metric dimension and dominance number. Moreover, we obtain the automorphism group Aut(Δ(Bn)) and the endomorphism monoid End(Δ(Bn)) of Δ(Bn). We show that Aut(Δ(Bn)) = S_n ×Z_2, where S_n is the symmetric group of degree n and Z_2 is the additive group of integers modulo 2. Further, for n ≥4, we prove that End(Δ(Bn)) =Aut(Δ(Bn)). In order to provide an answer to the question posed in [2], we ascertained a class of inverse semigroups whose commuting graph is Hamiltonian.
In this paper, we continue investigation of decompositions of complete graphs into graphs with seven edges. The spectrum has been completely determined for such graphs with at most six vertices. Connected graphs with seven edges and seven vertices are necessarily unicyclic and the spectrum for bipartite ones was completely determined by the authors. Connected graphs with seven edges and eight vertices are trees and the spectrum was found by Huang and Rosa. As a next step in the quest of completing the spectrum for all graphs with seven edges, we completely solve the case of disconnected bipartite graphs with seven edges and eight vertices.
A signed graph G(sigma)=(G,sigma) consists of an underlying graph G=(V,E) along with a signature function sigma:E ->{-1,1}. A cycle in a signed graph is termed positive if it contains an even number of negative edges, and negative if it contains an odd number of negative edges. A signed graph is considered { balanced} if it has no negative cycles; otherwise, it is { unbalanced}. Let K-m,K-n be a { complete bipartite graph} on m+n vertices. It is well known that for a balanced complete bipartite signed graph K-m,n(sigma), the parameters m and n are Laplacian eigenvalues with multiplicities n-1 and m-1, respectively. This raises a natural question about the maximum multiplicities of Laplacian eigenvalues m and n in an unbalanced complete bipartite signed graph Km,n sigma. In this paper, we demonstrate that the multiplicities of the Laplacian eigenvalues m and n in an unbalanced complete bipartite signed graph K-m,n(sigma) are at most n-2 and m-2, respectively. Additionally, we characterize all the signed graphs for which m and n are Laplacian eigenvalues with these maximum multiplicities.
Let Phi = (G, phi) be a T-gain (or complex unit gain) graph and A(Phi) be its adjacency matrix. The nullity of Phi, denoted by eta(Phi), is the multiplicity of zero as an eigenvalue of A(Phi), and the cyclomatic number of Phi is defined by c(Phi) = e(Phi) - n(Phi) + kappa(Phi), where n(Phi), e(Phi) and kappa(Phi) are the number of vertices, edges and connected components of Phi, respectively. A connected graph is said to be cycle-spliced if every block in it is a cycle. We consider the nullity of cycle-spliced T-gain graphs. Given a cycle-spliced T-gain graph Phi with c(Phi) cycles, we prove that 0 < eta(Phi) < c(Phi) + 1. Moreover, we show that there is no cycle-spliced T-gain graph Phi of any order with eta(Phi) = c(Phi) whenever there are no odd cycles whose gain has real part 0. We give examples of cycle-spliced T-gain graphs whose nullity equals the cyclomatic number, and we show some properties of those graphs Phi such that eta(Phi) = c(Phi) - epsilon, epsilon is an element of {0, 1 }. A characterization is given in case eta(Phi) = c(Phi) when Phi is obtained by identifying a unique common vertex of 2 cycle-spliced T-gain graphs Phi( 1) and Phi (2) . Finally, we compute the nullity of all T-gain graphs Phi with c(Phi) = 2.
The definition of the weighted topological index associated with a degree function phi is Phi(G) = Euv is an element of E (G) phi(du,dv), where du denotes the degree of node u and phi satisfies symmetric property phi(du, dv) = phi(dv, du). In this paper, we characterized extremal graphs and presented several results concerning the function Phi(G) in terms of various graph invariants. Additionally, we characterize the graphs that achieve these bounds and present multiple bounds for Phi(G) for the class of cozero divisor graphs defined on commutative rings.
