On Local-Strong Rainbow Connection Numbers On Generalized Prism Graphs And Generalized Antiprism Graphs

Rainbow geodesic is the shortest path that connects two different vertices in graph  such that every edge of the path has different colors. The strong rainbow connection number of a graph G, denoted by src(G), is the smallest number of colors required to color the edges of G such that there is a rainbow geodesic for each pair of vertices. The d-local strong rainbow connection number, denoted by lrscd, is the smallest number of colors required to color the edges of G such that any pair of vertices with a maximum distance d is connected by a rainbow geodesic. This paper contains some results of lrscd of  generalized prism graphs (PmxCn) and generalized antiprism graphs  for values of d=2, d=3, and d=4

Magic and antimagic labeling of graphs

"A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling."Doctor of Philosoph

CHARACTERISTIC ANTIADJACENCY MATRIX OF GRAPH JOIN

Let  be a simple, connected, and undirected graph. The graph  can be represented as a matrix such as antiadjacency matrix. An antiadjacency matrix for an undirected graph with order  is a matrix that has an order  and symmetric so that the antiadjacency matrix has a determinant and characteristic polynomial. In this paper, we discuss the properties of antiadjacency matrix of a graph join, such as its determinant and characteristic polynomial. A graph join  is obtained of a graph join operation obtained from joining two disjoint graphs  and

ON ANTIADJACENCY MATRIX OF A DIGRAPH WITH DIRECTED DIGON(S)

The antiadjacency matrix is one representation matrix of a digraph. In this paper, we find the determinant and the characteristic polynomial of the antiadjacency matrix of a digraph with directed digon(s). The digraph that we will discuss is a digraph obtained by adding arc(s) in an arborescence path digraph such that it contained directed digon(s), and a digraph obtained by deleting arc(s) in a complete star digraph. We found that the determinant and the coefficient of the characteristic polynomial of the antiadjacency matrix of a digraph obtained by adding arc(s) in an arborescence path digraph such that it contained directed digon(s) is different depending on the location of the directed digon. Meanwhile, the determinant of the antiadjacency matrix of a digraph obtained by deleting arc(s) in the complete star digraph is zero

Construction of super edge magic total graphs

E

Properties of consecutive edge magic total graphs

E

On Total Irregularity Strength of Double-Star and Related Graphs

AbstractLet G = (V, E) be a simple and undirected graph with a vertex set V and an 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 x1, their weights and are distinct, and for any two different edges xy and x1y1 their weights f (x) + f (xy) + f (y) and f (x1) + f (x1y1) + f (y1) are also distinct. A total irregularity strength of graph G, denoted byts(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 double-star S n,m, n, m ≥ 3 and graph related to it, that is a caterpillar S n,2,n, n ≥ 3. The results are and ts(S n,2,n) = n

Eigenvalues of antiadjacency matrix of Cayley graph of Z_n

In this paper, we give a relation between the eigenvalues of the antiadjacency matrix of Cay(Z_n, S) and the eigenvalues of the antiadjacency matrix of Cay(Z_n, (Z_n−{0})−S), as well as the eigenvalues of the adjacency matrix of Cay(Z_n, S). Then, we give the characterization of connection set S where the eigenvalues of the antiadjacency matrix of Cay(Z_n, S) are all integers.</p

Sum graph based access structure in a secret sharing scheme

Secret sharing scheme is a method to distribute secret information to a set P of participants so that only authorised subsets of P can reconstruct the secret. A set of subsets of P that can reconstruct the secret is called an access structure of the scheme. A simple undirected graph G is called a sum graph if there exists a labeling L of the vertices of G into distinct numbers, usually positive integers, such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label is L(w) = L(u) + L(v). In this paper, we will show how sum labeling can be used for representing the graphs of the access structures of a secret sharing scheme. We will combine a known secret sharing scheme such as the classical Shamir scheme with a graph access structure represented using sum graph labeling to obtain a new secret sharing scheme.C

Local Inclusive Distance Vertex Irregular Graphs

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight of a vertex x∈V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d=1 and determine the precise values of the corresponding graph invariant for certain families of graphs