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In this paper we determine the pair difference cordial number of degree splitting graph of bistar, complete bipartite graph, ladder, wheel. [ABSTRACT FROM AUTHOR]

In this paper, we obtain necessary and sufficient conditions for decomposing tensor product of complete graphs into some connected unicyclic bipartite graphs with eight edges. [ABSTRACT FROM AUTHOR]

Let G(V, E) be a graph with order p and size q. A bijection f : V (G) → {0, 1, 2, · · ·, p − 1} is said to be a cube sum labeling if the induced function f∗ : E(G) → N defined by f∗ (uv) = [f(u)]³ + [f(v)]³ is injective. Such a function f is said to be a cube sum labeling and the graph G is a cube sum graph. In this paper we discuss some algebraic properties and evaluate some families of cube sum graph. [ABSTRACT FROM AUTHOR]

Let G be a graph. Let f : V (G) → {0, 1, 2, ..., k -- 1} be a function where k ∈ ℕ and k > 1. For each edge uv, assign the label f (uv) = [f(u)+f(v)/2]. f is called k-total mean cordial labeling of G if |tmf (i) -- tmf (j)| ≤ 1 for all i, j ∈ {0, 1, 2, ⋯, k --1} where tmf (x) denotes the total number of vertices and edges labelled with x, x ∈ {0, 1, 2, ⋯, k -- 1}. A raph with admit a k-total mean cordial labeling is called k-total mean cordial graph. [ABSTRACT FROM AUTHOR]

In this paper, the new concept Topologized domination on Neutrosophic Graphs is introduced. The idea of N-Top domination is discussed in cycle, path, complete graph, star graph. The basic properties of N-Top dom set, N-Top minimum dom set, N-Top minimal dom set are introduced and N-Top dom number is also established with some necessary examples. [ABSTRACT FROM AUTHOR]

In this article, we have examined the Wiener index in neutrosophic graphs. Wiener index is one of the most important topological indices. This index is a distance-based index that is calculated based on the geodesic distance between two vertices. Here, after defining the Wiener index in neutrosophic graphs, we calculated this index for some special modes such as the complete neutrosophic graph, cycle, and tree. In the following, by presenting a several theorems, we compared this index with the connectivity index, which is one of the most important degree-based indicators. [ABSTRACT FROM AUTHOR]

In this paper, we define the Laplacian of product of Randić and sum-connectivity energy of a graph. Then, we compute the Laplacian of product of Randić and sumconnectivity energies of complete graph, star graph, complete bipartite graph, the crown graph, the (Sm ∧P2) graph. [ABSTRACT FROM AUTHOR]

Neutrosophic Graphs are graphs that follow three-valued logic. They may be considered a fuzzy graph, although in some cases, it is difficult to optimize and model them using fuzzy graphs. In this paper, the first and second Zagreb indices, the Harmonic index, the Randic' index and the Connectivity index for these graphs are investigated and some of the theorems related to these indices are discussed and proven. These indices are also calculated for some specific types of Neutrosophic Graphs, such as regular Neutrosophic Graphs and regular complete Neutrosophic Graphs. [ABSTRACT FROM AUTHOR]

In this paper we define the quotient of Randić and sum-connectivity energy of a graph. Then we obtain upper and lower bounds for Eqrs(G), quotient of Randić and sum-connectivity energy of a graph. Further we compute the quotient of Randić and sum-connectivity energies of complete graph, star graph, complete bipartite graph, the (Sm -P2) graph. [ABSTRACT FROM AUTHOR]

A barbell graph B(r, n) is a graph consists of path Pn joining two complete graphs Kr. This paper deals with study of the product cordial labeling of graphs that are obtained by applying various graph operations on barbell graph. [ABSTRACT FROM AUTHOR]

Let G be a (p, q) graph. Let f : V (G) → {1, 2, ..., k} be a map where k is an integer 2 ≤ k ≤ p. For each edge uv, assign the label |f(u) - f(v)|. f is called k-difference cordial labeling of G if |vf (i) - vf (j)| ≤ 1 and |ef (0) - ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x, ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate 3-difference cordial labeling behavior of DTn ⊙ K1 DTn ⊙ 2K1,DTn ⊙ K2 and some more graphs. [ABSTRACT FROM AUTHOR]

In this paper, the exact formulae for the generalized product degree dis-tance, reciprocal product degree distance and product degree distance of strong prod-uct of a connected graph and the complete multipartite graph with partite sets of sizes m0, m1, · · ·, mr-1 are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new concept named the uniform single valued neutrosophic graph. An illustrative example and some properties are examined. Next, we develop an algorithmic approach for computing the complement of the single valued neutrosophic graph. A numerical example is demonstrated for computing the complement of single valued neutrosophic graphs and uniform single valued neutrosophic graph. [ABSTRACT FROM AUTHOR]

A digraph D is eulerian if D is strongly connected and for every vertex v ∈ V (D), d+ (v) = d- (v). Bang-Jensen and Thomassé gave the conjecture for a digraph D that if λ(D) ≥ α(D), then D is supereulerian. Bang-Jensen and Maddaloni [Journal of Graph Theory, 79(2015)8-20] proved that this conjecture holds for every semicomplete multipartite digraph. In this paper, we generalized the above known results and show that this conjecture holds for every strong locally semicomplete multipartite digraph. [ABSTRACT FROM AUTHOR]

Let G be the dihedral group Dn and Cay(G, S) is the Cayley graph of G with respect to S, and let CG(x) is the centralizer of an element x in G and x is the orbit of x in G. In this paper, we prove that if G act on G by conjugation, the vertex induced subgraph with vertex set CG(x) of the Cayley graph Cay(G, x) is either Hamiltonian or it contain Hamiltonian decompositions. But if n is prime, it is always Hamiltonian. [ABSTRACT FROM AUTHOR]

All connected Cayley graphs over Abelian groups are Hamiltonian. However, for Cayley graphs over non-Abelian groups, Chen and Quimpo prove in [2] that Cayley graphs over Hamiltonian groups (i.e, non- Abelian groups in which every subgroup is normal) are Hamiltonian. In this paper we discuss a few of the ideas which have been developed to establish the existence of Hamiltonian cycles and paths in the vertex induced subgraphs of Cayley graphs over non-Abelian groups. [ABSTRACT FROM AUTHOR]

Let Fk(G),k ≥ 1, G be the token graph of a connected graph G. In this paper, we investigate the Eulerian and Hamiltonian property of token graphs and obtain the covering invariants for complete graph of token graph. [ABSTRACT FROM AUTHOR]

In this paper we introduce new graph labeling called k-difference cordial labeling. Let G be a (p,q) graph and k be an integer, 2 ≤ k ≤ |V(G)|. Let f : V(G) → {1, 2, • • •,k} be a map. For each edge uv, assign the label |f (u) − f (v)|. f is called a kdifference cordial labeling of G if |vf(i) − vf (j)| ≤ 1 and |ef(0) − ef (1)| ≤ 1 where vf (x) denote the number of vertices labelled with x, ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate k-difference cordial labeling behavior of star, m copies of star and we prove that every graph is a subgraph of a connected k-difference cordial graph. Also we investigate 3-difference cordial labeling behavior of some graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of graph labeling known as total mean cordial labeling. A total mean cordial labeling of a graph G = (V,E) is a mapping f : V (G) → {0, 1, 2} such that f(xy) = ... where x, y ∈ V (G), xy ∈ G, and the total number of 0, 1 and 2 are balanced. That is |evf (i) − evf (j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). If there exists a total mean cordial labeling on a graph G, we will call G is Total Mean Cordial. In this paper, we study some classes of graphs and their Total Mean Cordial behaviour. [ABSTRACT FROM AUTHOR]

In this paper, we find the harmonious chromatic number on Mycielskian graph of cycle, path, complete graph and complete bipartite graph. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be an integer. The complete graph Kn with 1-factor I added has a decomposition into Hamilton cycles if and only if n is even. We show that Kn + I has a decomposition into Hamilton cycles which are symmetric with respect to the 1-factor I added. We also show that the complete bipartite graph Kn,n plus a 1-factor has a symmetric Hamilton decomposition, where n is odd. [ABSTRACT FROM AUTHOR]

The sandpile group or critical group of a graph is an Abelian group whose order is the number of spanning trees of the graph. In the paper, the order of the sandpile group of infinite complete expansion regular graphs is obtained. [ABSTRACT FROM AUTHOR]

A divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, ⋯ , |V |} such that an edge uv is assigned the label 1 if f(u) divides f(v) or f(v) divides f(u) and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. If a graph has a divisor cordial labeling, then it is called divisor cordial graph. In this paper, we proved the standard graphs such as path, cycle, wheel, star and some complete bipartite graphs are divisor cordial. We also proved that complete graph is not divisor cordial. [ABSTRACT FROM AUTHOR]

In this paper, we provide three new operations on fuzzy graphs; namely direct product, semi-strong product and strong product. We give sufficient conditions for each one of them to be complete and we show that if any of these products is complete, then at least one factor is a complete fuzzy graph. Moreover, we introduce and study the notion of balanced fuzzzy graph and give necessary and sufficient conditions for the preceding products of two fuzzy balanced graphs to be balanced and we prove that any isomorphic fuzzy graph to a balanced fuzzy graph must be balanced. [ABSTRACT FROM AUTHOR]

For two vertices u and v in a graph G = (V,E), the distance d(u, v) and detour distance D(u, v) are the length of a shortest or longest u - v path in G, respectively, and the Smarandache distance di S(u, v) is the length d(u, v)+ i(u, v) of a u-v path in G, where 0 ⩽ i(u, v) ⩽ D(u, v) - d(u, v). A u - v path of length di S(u, v), if it exists, is called a Smarandachely u - v i-detour. A set S ⊆ V is called a Smarandachely i-detour set if every edge in G has both its ends in S or it lies on a Smarandachely i-detour joining a pair of vertices in S. In particular, if i(u, v) = 0, then di S(u, v) = d(u, v); and if i(u, v) = D(u, v) - d(u, v), then di S(u, v) = D(u, v). For i(u, v) = D(u, v) - d(u, v), such a Smarandachely i-detour set is called a weak edge detour set in G. The weak edge detour number dnw(G) of G is the minimum order of its weak edge detour sets and any weak edge detour set of order dnw(G) is a weak edge detour basis of G. For any weak edge detour basis S of G, a subset T ⊆ S is called a forcing subset for S if S is the unique weak edge detour basis containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing weak edge detour number of S, denoted by fdnw(S), is the cardinality of a minimum forcing subset for S. The forcing weak edge detour number of G, denoted by fdnw(G), is fdnw(G) = min(fdnw(S)), where the minimum is taken over all weak edge detour bases S in G. The forcing weak edge detour numbers of certain classes of graphs are determined. It is proved that for each pair a, b of integers with 0 ⩽ a ⩽ b and b ⩾ 2, there is a connected graph G with fdnw(G) = a and dnw(G) = b. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 1, a cycle-complete graph Smarandache-Ramsey number rsk (Cm,Kn) is the smallest integer N such that every graph G of order N contains k cycles, Cm, on m vertices or the complement of G contains k complete graph, Kn, on n vertices. If k = 1, then the Smarandache-Ramsey number rsk (Cm,Kn) is nothing but the classical Ramsey number r(Cm,Kn). Radziszowski and Tse proved that r(C4,K9) ≥ 30. Also, By considering the known graph G = 7K4, we have that r(C5,K8) ≥ 29. In this paper we give an upper bound of r(C4,K9) and r(C5,K8). [ABSTRACT FROM AUTHOR]