Search

Consider a graph G with a nonempty set of vertices V (G) and a set of edges E(G). Let Zn represent the group of integers modulo n, and let k be a positive integer. A modular irregular labeling of a graph G with order n is a labeling of its edges, φ : E(G) → {1, 2, ..., k}, such that a weight function σ : V (G) → Zn is induced. The weight function is defined as follows: σ (v) = ∑u∈N(v) φ(uv) for all vertices v in V (G), where the summation is taken over all vertices u that adjacent with vertex v in G, and this weight function σ must be bijective. The minimum value of k such that a labeling exists in a graph G is called the modular irregularity strength of G, denoted as ms(G). In this research, we have determined the exact values of the modular irregularity strength for generalized book graphs. [ABSTRACT FROM AUTHOR]

Adjacency and antiadjacency matrices are the representation matrices of a directed graph. From the earlier papers, its have been known that the powering of the adjacency matrix of a directed graph can be used to find the number of directed paths and cycles. However, we found that there is a case in which the theorem does not work. Therefore, in this paper, we will extend the powering of the adjacency matrix of a directed graph and also give the additional requirement so that the properties are worked. We also generalized the case for a general directed graph, which means the graph might have loop(s) and digon(s). On the other hand, we also give the representation of the powering of the antiadjacency matrix. [ABSTRACT FROM AUTHOR]

LetG = (V, E) be a simple and undirected graph with |V| vertices and |E| edges. Consider a graph G with |E| ≥ |V|. An injective f from V to a set {0,1,2,... , |E|−1} such that the induced edge labeling given by f(xy) = g(x) + g(y) (mod |E|) for any edge xy in the graph is also an injective function, is called harmonious labeling of a graph G. A harmonious graph is a graph which has a harmonious labeling. In this paper we show an existence of harmonious labeling on G + K ¯ 2 and G × P2, where G is a harmonious unicyclic graph. [ABSTRACT FROM AUTHOR]

A graph is a set of vertices and edges where each edge connects two vertices in the graph. There are several ways to represent a graph, e.g., it can be represented as a matrix: its adjacency matrix, anti-adjacency matrix, Laplacian matrix, or distance matrix. Two matrices which will be explored in this paper are the adjacency matrix and distance matrix. The adjacency matrix represents the presence or absence of an edge connecting two vertices, while the distance matrix represents the shortest path between two vertices. A graph with diameter two is a graph such that the longest distance between any two vertices is equal to two. Examples of two-diameter graphs include bipartite graphs, wheel graphs, and fan graphs. The relation between the distance and adjacency matrices is already known. The purpose of this paper is to find the relation between the square of the distance matrix and the square of the adjacency matrix of a two diameter graph and to find the characteristic polynomial of the distance matrix of a complete bipartite graph Kn,n(which is a special type of two-diameter graph). We prove that D² = 4(J-I)Ā + A2 where D is the distance matrix, J is the matrix all of whose entries are 1, I is the identity matrix, A is the adjacency matrix and Ā is the complement of A Moreover, we also find that the characteristic polynomial of the distance matrix of a complete p bipartite graph is P(λ) = (λ+2)2n-2(λ-(n-2))(λ-(3n-2)). [ABSTRACT FROM AUTHOR]

For a simple and connected graph G, the distance between two vertices u, v ϵ V (G), denoted by d(u, v), is the length of the shortest path connecting them. A vertex labeling λ: V (G) --> {1, …, k} is called an inclusive vertex irregular d-distance labeling if the weights of the vertices are distinct, where the weight of a vertex v is defined as the sum of the vertex label v and all vertex labels u such that d(u, v) ≤ d. The minimal value of the largest label k of all such labeling of G is called the inclusive d-distance irregularity strength of G and is denoted by dis0d (G). Lower and upper bounds for dis0d (G) have already been investigated for any graph G with d = 1. The value of dis01 (G) for some classes of graphs, such as paths, cycles, complete graphs, and other related types of graphs, have also been determined in the literature. In this paper, we find an upper bound for dis01 (G) for triangular ladder graphs Ln, with n ≡ 4 mod 5, n > 4, and the exact value for the rest of the cases with n ≥ 3. [ABSTRACT FROM AUTHOR]

Let G be a graph with V(G) = {v1,...,vn} and E(G) = {e1,...,.em}. We only consider graphs with no multiple edges in this paper. The adjacency matrix of G, denoted by A(G), is the n × n matrix A = [aij], where aij = 1 if e = vivj ϵ E(G) or aij = 0 otherwise. The antiadjacency matrix of G, denoted by B(G), is the n ×n matrix B = [bij], where bij = 0 if e = vivj ϵ E (G) or Bij = 1 otherwise. Harary and Wilcox have considered Boolean operations on two graphs G1 and G2, resulting a new graph G whose V(G1) × V (G) equals V(G1) × V(G2). On this paper, Boolean operations are defined for two adjacency and two antiadjacency matrices of graphs G1 and G2 with V(G1) = V(G2) rather than looking at the two graphs themselves. Boolean operations which are reviewed on this paper are V and Λ. The objectives of this paper are to introduce the operations on two adjacency or two antiadjacency matrices of graph G, to discover some characteristics of the operations on the matrices, to construct a new graph which is generated using the operators on two adjacency or antiadjacency matrices, and to reveal the similarity between operators on adjacency matrix and operators on antiadjacency matrix based on the represented graph. This paper also emphasizes on investigating the relationship between the operators and on comparing the largest eigenvalue between graphs which are constructed by Boolean operators on both adjacency and antiadjacency matrices. [ABSTRACT FROM AUTHOR]

Let G be a graph with V(G) = {v1,...,vn} and E(G) = {e1,...,.em}. In this paper, only graphs with no multiple edges are observed. The n × n matrix A = [aij}, where aij = 1 if e = v1vj and aij = 0 otherwise, is the adjacency matrix of G and is denoted by A(G). The n × n matrix B = [bij], where bij = 0 if e = vivj ϵ E (G) and bij = 1 otherwise, is the antiadjacency matrix of G and is denoted by B(G). Boolean operations on two graphs have been examined by Harary and Wilcox. Hence, this paper will consider Boolean operations on two adjancency and antiadjancency matrices of two graphs G1 and G2 with V(G2) = V(G2). Boolean operations which are reviewed on this paper are OR (V), AND (Λ), XOR(⊕), and NXOR(⊕) However, the paper only focus on operation ⊕ and ⊕. The purposes of this paper are to introduce the operations on two adjacency and antiadjacency matrices of two graphs G1 and G2 with V(G1) = V(G2), to reveal the effect on the represented graph using operations ⊕ and ⊕ on both adjacency and antiadjacency matrices, and to compare the largest eigenvalues between the matrices generated by the Boolean operations. [ABSTRACT FROM AUTHOR]

A graph is called an edge uncertain graph if the edges are not deterministic but exist with some belief degrees described by uncertain measure. A new definition for total uncertain graph is first introduced in this paper, that is an uncertain graph in which the vertices and the edges are not deterministic. Further, we give a concept of an α-cut of a total uncertain graph and investigate some of its properties. We color a total uncertain graph via the α-cut, and obtain α-cut chromatic numbers for all α ∈ [0,1]. Some properties of the α-cut chromatic number are also verified. [ABSTRACT FROM AUTHOR]

For a simple graph G = (V,E) with the vertex set V and the edge set E, a totally irregular total k-labeling f : V ∪ E → {1,2, . . ., k} is a labeling of vertices and edges of G in such a way that for any two different vertices x and x', their weights wt f (x) = f (x)+Σxy∈E f(xy) and wtf(x') = f (x')+Σx'y'∈E f(x'y') are distinct, and for any two different edges xy and x'y' their weights f (x)+ f(xy)+ f(y) and f(x')+ f(x'y')+ f(y') are also distinct. A total irregularity strength of graph G, denoted by ts(G), is defined as the minimum k for which G has a totally irregular total k-labeling. In this paper, we determine the exact value of the total irregularity strength for star graphs, double stars and caterpillar. [ABSTRACT FROM AUTHOR]