Product Cordial Graphs Research Articles

On bijective product square k-cordial labeling

A bijective product square [Formula: see text]-cordial labeling of a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text] is a bijection [Formula: see text] from [Formula: see text] to [Formula: see text] such that the induced edge function [Formula: see text] from [Formula: see text] to [Formula: see text] defined as [Formula: see text] for every edge [Formula: see text] satisfies the following condition. (i) If [Formula: see text] is the number of edges labeled with [Formula: see text] under [Formula: see text], then [Formula: see text] where [Formula: see text]. A graph which admits a bijective product square [Formula: see text]-cordial labeling is called bijective product square [Formula: see text]-cordial graph. In this paper, we find upper limit of the size of the bijective product square [Formula: see text]-cordial graph for [Formula: see text] and [Formula: see text]. Also, we prove that every graph is a subgraph of a connected bijective product square [Formula: see text]-cordial graph for all odd prime [Formula: see text]. In addition, we establish the relation between bijective product square [Formula: see text]-cordial and [Formula: see text]-product cordial graphs.

ENERGY OF FIBONACCI PRODUCT CORDIAL GRAPH

Let G be a Fibonacci product cordial graph with n vertices. Then the Fibonacci product cordial label energy of G is denoted by EFPCL [G] and is defined as EFPCL[G]= n i=1 |ρi|, where ρi are the eigenvalues of the Fibonacci product cordial labeled matrix. In this paper, we introduce the Fibonacci product cordial labeled matrix, Fibonacci Product Cordial Labeled Laplacian matrix, Fibonacci product cordial label Laplacian energy, Fibonacci product cordial label equi-laplacian energic graphs respectively.

Product Cordial Labelling for Some Bicyclic Graphs

A graph \(G\) with lines and points is known as a product cordial graph if there occurs a labeling \(g\) from \(V(G)\) to \(\{0,1\}\) such that if every line \(rt\) is given the labeled \(g(r) .g(t)\), then the cardinality of points with labeled zero and the cardinality of points with labeled one vary as a maximum by one and the cardinality of lines with labeled zero and the cardinality of lines with labeled one vary by as a maximum one. In this case, \(g\) is alleged the product cordial labeling of \(G\). This paper deals with product cordial labeling for some graphs related to bicyclic graph such as \(B[n,n]\), \(B[n,n]*S_m\), \(B[n,n]*P_2*S_m\) and \(B[n,n]*P_3*S_m\), \(B[n,n] \odot K_2\), \(B[n,n] \odot K_3\).

Product cordial graph in the context of some graph operations on crown, helm, and wheel graph

Product cordial graph in the context of some graph operations on crown, helm, and wheel graph

On balanced 3-edge product cordial graphs

A k-edge product cordial labelling is a variant of the well-known cordial labelling. In this paper, a balanced k-edge product cordial labelling is suggested and some sufficient conditions for balanced 3-edge product cordial graphs are proved. Moreover, a construction of graphs admitting a balanced 3-edge product cordial labelling is presented.

On balanced edge product cordial graphs

An edge product cordial labelling is a variant of the well-known cordial labelling. In this paper, a balanced edge product cordial labelling is suggested and some sufficient conditions for balanced edge product cordial graphs are proved. Also, a construction of balanced edge product cordial graphs is presented.

On 3-total edge product cordial labeling of grid

Let us consider a mapping [Formula: see text] of a graph [Formula: see text], where [Formula: see text] is an integer, [Formula: see text]. The mapping [Formula: see text] induces for every vertex [Formula: see text] of [Formula: see text] the label [Formula: see text]. Let [Formula: see text] ([Formula: see text]) denote the number of edges (vertices) in [Formula: see text] that are labeled with the number [Formula: see text] under the labeling [Formula: see text], [Formula: see text].The function [Formula: see text] is called a [Formula: see text]-total edge product cordial labeling of [Formula: see text] if [Formula: see text] for [Formula: see text]. A graph [Formula: see text] with a [Formula: see text]-total edge product cordial labeling is called a [Formula: see text]-total edge product cordial graph.In this paper, we prove that the grid graph [Formula: see text] for [Formula: see text] admits a [Formula: see text]-total edge product cordial labeling.

On edge product cordial graphs

An edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize graphs admitting an edge product cordial labeling. Using this characterization we investigate the edge product cordiality of broad classes of graphs, namely, dense graphs, dense bipartite graphs, connected regular graphs, unions of some graphs, direct products of some bipartite graphs, joins of some graphs, maximal \(k\)-degenerate and related graphs, product cordial graphs.

Cycle Related Near and Weak Product Cordial Graph

Cycle Related Near and Weak Product Cordial Graph

Product Cordial Graph in the Context of Some Graph Operations on Gear Graph

A graph is said to be a product cordial graph if there exists a function with each edge assign the label , such that the number of vertices with label 0 and the number of vertices with label 1 differ atmost by 1, and the number of edges with label 0 and the number of edges with label 1 differ by atmost 1. We discuss the product cordial labeling of the graphs obtained by duplication of some graph elements of gear graph. Also, we derive some product cordial graphs obtained by vertex switching operation on gear graph.

Some Edge Product Cordial Graphs in the Context of Duplication of Some Graph Elements

For a graph, a function is called an edge product cordial labeling of G, if the induced vertex labeling function is defined by the product of the labels of the incident edges as such that the number of edges with label 1 and the number of edges with label 0 differ by at most 1 and the number of vertices with label 1 and the number of vertices with label 0 differ by at most 1. In this paper, we show that the graphs obtained by duplication of a vertex, duplication of a vertex by an edge or duplication of an edge by a vertex in a crown graph are edge product cordial. Moreover, we show that the graph obtained by duplication of each of the vertices of degree three by an edge in a gear graph is edge product cordial. We also show that the graph obtained by duplication of each of the pendent vertices by a new vertex in a helm graph is edge product cordial.

Some New Classes of 3-Total Product Cordial Graphs

In [ 3 ], Ponraj. R et al have defined the 3- Total Product Cordial of a graph G V, E as follows, Let f be a function from VG to{0, 1, ... k - 1}where k is an integer, 2 ≤ k ≤ VG . For each edge uv assign the label f u f v mod k. f is called a k - Total Product cordial labeling if f i − f j ≤ 1, i, j ∈ {0, 1, . .k - 1} where fx denotes the total number of vertices and edges labeled with xx = 0, 1, 2, ...., k-1. We prove that the 3-Total Product cordial labeling is a behaviour of Fn.

PRODUCT CORDIAL LABELING OF GRAPHS RELATED TO HELM, CLOSED HELM AND GEAR GRAPH

We prove that path union of finite copies of helm, closed helm, gear graph are product cordial graphs. Further we prove that the graph obtained by joining two copies of helm by a path of arbitrary length is product cordial graph. We prove similar results for closed helm and gear graph.

Some Edge Product Cordial Graphs

For a graph $G=(V(G), E(G))$, a function $ f : E(G) \rightarrow \{0, 1\}$ is called an edge product cordial labeling of $G$ if the induced vertex labeling function defined by the product of incident edge labels be such that the edges with label 1 and label 0 differ by at most 1 and the vertices with label 1 and label 0 also differ by at most 1. In this paper we investigate some new families of edge product cordial graph.

Further Results on Product Cordial Graphs

A binary vertex labeling of graph $ G $ with induced edge labeling $ f^{*}:E(G) \rightarrow \{0,1\} $ defined by $ f^{*}(e=uv)=f(u)f(v) $ is called a \textit{product cordial labeling} if $ \vert v_{f}(0) - v_{f}(1) \vert \leq 1 $ and $ \vert e_{f}(0) - e_{f}(1) \vert \leq 1 $. A graph is called \textit{product cordial} if it admits product cordial labeling. We prove that the shell admits a product cordial labeling. Sundaram et al.\cite{Sundaram1} proved that if a graph with $p$ vertices and $ q$ edges with $ p \geq 4 $ is product cordial then $ q \leq \dfrac{p^{2}-1}{4} + 1 $. We present here some families of graphs which satisfy this condition but not product cordial.

Product Cordial Labeling in the Context of Tensor Product of Graphs

For the graph $G_{1}$ and $G_{2}$ the tensor product is denoted by $G_{1}(T_{p})G_{2}$ which is the graph with vertex set $V(G_{1}(T_{p})G_{2}) = V(G_{1}) \times V(G_{2})$ and edge set $E(G_{1}(T_{p})G_{2})= \{(u_{1},v_{1}),(u_{2},v_{2})/u_{1}u_{2} \epsilon E(G_{1})$ and $v_{1}v_{2} \epsilon E(G_{2})\}$. The graph $P_{m}(T_{p})P_{n}$ is disconnected for $\forall m,n$ while the graphs $C_{m}(T_{p})C_{n}$ and $C_{m}(T_{p})P_{n}$ are disconnected for both $m$ and $n$ even. We prove that these graphs are product cordial graphs. In addition to this we show that the graphs obtained by joining the connected components of respective graphs by a path of arbitrary length also admit product cordial labeling.

Product Cordial Labeling for Some New Graphs

In this paper we investigate product cordial labeling for some new graphs. We prove that the friendship graph, cycle with one chord (except when n is even and the chord joining the vertices at diameter distance), cycle with twin chords (except when n is even and one of the chord joining the vertices at diameter distance) are product cordial graphs. We also investigated middle graph of path $P_{n}$ admits product cordial labeling.