A Study of Cordial Labeling of Friendly Index Related Graphs

the friendly index set of graphs. Let G be a graph with vertex set V(G) and edge set E(G). A labeling f: V(G) → Z2 induces an edge labeling f*: E(G) → Z2 defined by f*(xy) = f(x) +f(y), for each edge xy Œ E(G). For i א Z2, let vf(i) = number of vertices v א V(G) with f(v) = i and ef(i) = number of edges e א E(G) with f*(e) = i. A labeling f of a graph G is said to be friendly if |vf(0) - vf(1)| ≤ 1. The friendly index set of the graph G, FI(G) is defined as {|ef(0) - ef(1)|: the vertex labeling f is friendly}. The following works have been done by several authors. Friendly index set of some graphs which are not dealt with is proved in this study.

Theorem 1: The friendly index set of one point union of t copies where t ≥ 5 and t is odd, of cycle Proof: Let the common vertex be w and a1, a2, ..., a2t be the vertices of t cycles. Make the w label equal to zero. Starting at 0, label the vertices with successively higher numbers. To rephrase: (a1, a2,..., a2t) = (0, 1, 0, 1, ...., 0, 1). In this case, vf(1) - vf(0) = 0 indicates a pleasant vertex labeling. Without altering any other vertex labels, we rearrange the vertices by changing the labels a4i-2 and a4i-1, where I = 1, to their complement. The Similarly, with all the vertex unchanged and changing the vertex label a4i-2 and a4i-1, where i = 2 to its complement and is repeated up to Theorem 2: Union of the cycle C3 and a path Pn sharing a vertex in common has the friendly index set {1, 3, 5, …, n}, if n is odd and {0, 2, 4, …, n}, if n is even. Proof: Let u1, u2, u3 be the vertices of C3. Let w1, w2,…, wn be the vertices of Pn. Let w1 is identified with u1. Then the graph obtained is union of C3 and Pn sharing a vertex in common. Let it be G. Then keep the labels of w1 and w2 unchange and change all the remaining labels of the vertices to their complement. Then the vertex labeling is vf(1) - vf(0) = 1 and Then keep the labels of w1, w2, w3, w4 remain unchanged and change all the remaining vertices to its complement. Then the vertex labeling is friendly with Continuing this to the end of the path, we get one step That is, Finally change the label of wn to its complement and the vertex labeling is friendly and Then the friendly index set is {1, 3, 5, ..., n}

Theorem 1: FI(P2 + mk1) = {1, 3}, if m is even and FI(P2 + mk1) = {1, 5}, if m is odd. Proof : Consider a path P2 with two vertices v1, v2. Let y1, y2, ..., ym be the m isolated vertices. Join v1, v2 with yi, 1 ≤ i ≤ m. The graph obtained is P2 + mk1. The vertex set of G is V(G) = {v1, v2, y1, y2, ..., ym}. The edge set of G is E(G) = {(v1v2 )} » {(v1yi) | l ≤ i ≤ m} » {(v2yi) | l ≤ i ≤ m}. Note that the graph has m + 2 vertices and 2m + 1 edges Then label v1 as 0 and v2 as 1 and label the vertices of yi‘s as alternatively 0 and 1. This vertex labeling is also friendly with Therefore the friendly index set is {1, 3}. Theorem 2: Union of a path Pn and a star K1,n sharing a vertex in common has the friendly index set { 1, 3, 5, …, 2n - 5} Proof: Let the n vertices of path Pn be u1, u2, ..., un. Let the n spokes of star K1,n be v1, v2, …, vn and v0 be the centre vertex of the star. Identify u1 and v0. Let the graph so obtained is G. Now we have a friendly labeling by keeping all the vertex labels the same except for u2 and v2, which have been given their complements. Then Now, a pleasant labeling is accomplished by keeping all the vertex labels the same except for the fact that vertices u3 and v3 have been swapped. Then Continuing this process up to changing the vertex un-2 and vn-2 to its complement. Then the friendly index set is {1, 3, 5, …, 2n - 5} Theorem 3: The Umbrella graph U(m, n) [1.2.17] where n = 2 has the friendly index set {0, 2, 4, ..., m + 1}, if m is odd and m ≥ 9 and {0, 2, 4, ..., m}, if m is even and m ≥ 4 Note that the graph has m + 2 vertices and 2m edges Case (i) : m is odd First label alternatively with 0‘s and 1‘s starting with 0 in xi‘s and label y1 by 1 and y2 by 0. Then

After x1, the following two vertices are both labeled with 1, and the following two vertices are both labeled with 0, and so on. In that case, we can indicate that y1 is 1 and y2 is 0 by using those labels. Then

Theorem 1: The friendly index set of spl(k1,n) is {0, 2, 4, ..., n}, if n is even and {1, 3, ..., n}, if n is odd. Proof: Let v1, v2, ..., vn be the pendant vertices, v be the apex vertex of K1,n and u, u1, u2, .., un are the added vertices corresponding to v, v1, v2, ..., vn in spl (k1,n)

DIFFERENCE CORDIAL LABELING OF GRAPHS CYCLE

A concept called difference cordial labeling is presented in [45] by R. Ponraj, S. Sathish Narayanan, and R. Kala in 2013. In this case, we'll pretend that G is a (p, q) graph. Take into account the injective function f: V(G) [-1, +2..., +p]. With this, we obtain the map f* on E(G) with the equation f*(uv) = |f(u) - f(v)| for all u, v. This value is 1 if the difference is 1, and 0 otherwise. With ef(i) equal to the number of edges labeled with I where I = 0, 1, f is a difference cordial labeling if and only if |ef(0) - ef(1)| > 1. Differentially labeled graphs are referred to as "difference cordial graphs." These works are the product of a collaborative effort by multiple authors. v Any path is a difference cordial graph. v The star K1,n is difference cordial iff n ≤ 5. v Kn is difference cordial iff n ≤ 4. v B1,n is difference cordial iff n ≤ 5 v B2,n is difference cordial iff n ≤ 6 v B3,n is difference cordial iff n ≤ 5 v The gear graph Gn is difference cordial. v All webs are difference cordial v Every ladder graph Ln is difference cordial for all n. v The triangular ladder graph TLn, n ≥ 2 is a difference cordial for all n. v Every step ladder graph is a difference cordial for all n. v All Bow graphs are difference cordial. v All Butterfly graphs are difference cordial Now we shall prove the following graphs are difference cordial graphs. Theorem 1: A vertex switching of cycle Cn(VSCn) [1.2.22] is difference cordial, for all n ≥ 4 Proof: Let G be a (p, q) graph. A cycle's vertex switch is indicated by the symbol VSCn. One obtains it by starting with a vertex a1 in Cn, eliminating any edges incident with a1, and then adding edges attaching a1 to every vertices that is not adjacent to a1 in Cn. Illustration: Difference cordial labeling of VSC9 is given in figure 1

G is a (9, 13) graph Here number of edges labeled 0 is 6 That is, ef(0) = 6 Number of edges labeled 1 is 7 That is, ef(1) = 7 Then |ef(0) - ef(1)| = 1 Hence the above graph is difference cordial.

CONCLUSION

Labeling graphs is one of the most fascinating issues in Graph Theory. The assignment of labels to the edges (vertices) of a simple graph G is induced by a mapping from the vertex set (edge set) to a set of non-negative integers. Labeled graphs can be used as models in many contexts. They have several applications in coding theory, including the creation of high-quality radar-type codes. Best non-standard integer encodings in synch-set codes, missile guidance codes, and convolution codes. Labeled graphs have also been used to solve difficulties in radio astronomy, X-ray crystallography, and the design of a communication network's addressing scheme, among other areas of study. It was G Ringel's conjecture, "All trees are labeling. Graceful labeling enabled a great deal of productivity. The families have been elegantly numbered in the thousands. The definition and development of other labels occurred alongside the elegant labeling. It was Graham and Sloane who first used a graph with symmetry and harmony. Other names for this process include: the Prime label, the Arithmetic label, the Edge graceful label, the Felicitous label, the Antimagic label, the Cordial label, the Prime cordial label, the Edge cordial label, the Ek - cordial label, the Difference cordial label, the Friendly label, the Divisor cordial label, the Skolem graceful label, and the Product cordial label.

REFERENCES

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Research Scholar, Shri Krishna University, Chhatarpur, M.P.