Non-Adjacent Vertex Sum Polynomial of Umbrella Graphs, Jahangir Graph, Tadpole Graph and Lollipop Graph

Malaya Journal of Matematik, Vol. 9, No. 1, 941-944, 2021

https://doi.org/10.26637/MJM0901/0168

Non-adjacent vertex sum polynomial of Umbrella graphs, Jahangir graph, Tadpole graph and Lollipop graph

V. Jeba Rani1*, S. Sundar Raj2 and T. Shyla Isaac Mary3

Abstract Let G = (V,E) be a graph. The non-adjacent vertex sum polynomial of the graph G = (V,E) is defined as 4(G) α4(G)− j NAVSP(G,x) = ∑ j=0 n(4(G)− j)x where n4(G)− j is the sum of the number of non-adjacent vertices of all the vertices of degree 4(G) − j and α4(G)− j is the sum of the degree of non-adjacent vertices of the vertices of degree 4(G) − j. In this paper we derived the non-adjacent vertex sum polynomial for Umbrella graph, Jahangir graph, Tadpole graph and Lollipop graph. Keywords Umbrella graph, Jahangir graph, Tadpole graph and Lollipop graph. AMS Subject Classification 05C31. 1,3Department of Mathematics, Nesamony Memorial Christian College, Marthandam-629165, India. 2Department of Mathematics, Vivekananda College, Agasteeswaram–629203, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India *Corresponding author: 1 [email protected], 2 [email protected], 3 [email protected] Article History: Received 24 January 2021; Accepted 19 March 2021 c 2021 MJM.

Contents 2. Main Results

1 Introduction...... 941 Theorem 2.1. Let Um,n be a Umbrella graph with m + n ver- 2 Main Results...... 941 tices. Then the non-adjacent vertex sum polynomial of Um,n is NAVSP(U ,x) = References...... 944 m,n  6n−7 6n−9 2 6n−10 (2n − 2)x + (2n − 3)x + (2n − 11n + 12)x  5n−9 5n−10 5n−13 +(2n − 3)x + (4n − 6)x + (4n − 8)x + 1. Introduction  (2n2 − 12n + 16)x5n−14 + (n − 2)x2n−5  In a graph G = (V,E) we mean a finite undirected, non-trivial  i f m,n(m ≥ 5,n ≥ 5) are odd and graph without loops and multiple edges. The vertex set is  i f m,n(m ≥ 6,n ≥ 6) are even V E v ∈ V d(v)  denoted by and the edge set by , for , is the  6m−5 6m−7 (m + n − 2)x + (m + n − 3)x + number of edges incident with v, the maximum degree of G  (2n2 − 12n + 16)x6m−8 + (m + n − 3)x5m−7 is defined as 4(G) = max{d(v)/v ∈ V} for terms not defined  +(4n − 8)x5m−8 + (4n − 10)x5m−11+ here, we refer to Frank Harary [3]. An Umbrella graph Um,n  2 5m−12 2m−3 is the graph obtained by joining a path Pn with a central vertex (2n − 15n + 25)x + (n − 2)x  of a fan Fm.[7]. Jahangir graph Js,m for m ≥ 3 is a graph i f m(m ≥ 6) is even and i f n(n ≥ 7) is odd  on sm + 1 vertices that is a graph consisting of a cycle C (m + n − )x6m−9 + (m + n − )x6m−11+ sm  2 3  2 6m−12 5m−11 with one additional vertex which is adjacent to m vertices (2n − 10n + 8)x + (m + n − 3)x 0 0  of Csm at a distance s to each other an Csm [6]. A Tadpole +(4n − 4)x5m−12 + (4n − 6)x5m−15+  Tn,k is the graph obtained by appending a path Pk to cycle Cn  2 5m−16 2m−7 (2n − 9n + 9)x + (n − 2)x [7]. A Lollipop graph denoted by L is a graph which is  m,n i f m(m ≥ 7) is odd and i f n(n ≥ 6) is even constructed by appending a complete graph Km, m ≥ 3 to a pendent vertex of path graph Pn. Proof. Let V = {{ui,v j/1 ≤ i ≤ m,1 ≤ j ≤ n} be the vertex Non-adjacent vertex sum polynomial of Umbrella graphs, Jahangir graph, Tadpole graph and Lollipop graph — 942/944 set and E = {uiui+1,v jv j+1/1 ≤ i ≤ m − 1,1 ≤ j ≤ n − 1} ∪ {v1ui/1 ≤ i ≤ m} be the edge set of the graph Um,n. Then Um,n has m + n vertices and 2m + n − 2 edges.

Case(i) Both m (m ≥ 5) and n (n ≥ 5) are odd.

The total number of non-adjacent vertices of vn is 2n−2, vn−1 2 is 2n − 3,v3,v4,...,vn−2 is 2n -11n+12, v2 is 2n − 3, u1,um is 2 4n − 6, u2, um−1 is 4n-8,u3,u4,...,um−2 is 2n − 12n + 16 and v1 is n −2 and sum of degree of these non-adjacent vertices is 6n − 7,6n − 9,6n − 10,5n − 9,5n − 10,5n − 13,5n − 14 and Here NAVSP(U6,7,x) = 2n − 5 respectively. 11x31 +10x29 +30x28 +10x23 +20x22 +18x19 +18x18 +5x9.

Case (ii) Both m (m ≥ 6) and n (n ≥ 6) are even.

The result is similar to case (i).

Case (iii) m (m ≥ 6) is even and n (n ≥ 7) is odd.

The total number of non-adjacent vertices of vn is m + n − 2 2, vn−1 is m + n − 3,v3,v4,...,vn−2 is 2n − 12n + 16, v2 is m+n−3,u1,um is 4n−8,u2,um−1 is 4n−10,u3,u4,...,um−2 2 is 2n − 15 + 25 and v1 is n − 2 and sum of degree of these non-adjacent vertices is 6m − 5,6m − 7,6m − 8,5m − 7,5m − Here NAVSP(U7,6,x) = 8,5m − 11,5m − 12 and 2m − 3 respectively. 11x33 +10x31 +20x30 +10x24 +20x23 +18x20 +27x19 +4x7.

Case (iv) m (m ≥ 7) is odd and (n ≥ 6) is even. Theorem 2.3. Let Jsm be a Jahangir graph with sm + 1 vertices. Then the non-adjacent vertex sum polynomial of Jsm The total number of non-adjacent vertices of vn is m + n − is 2 2, vn−1 is m + n − 3, v3,v4,...,vn−2 is 2n − 10n + 8, v2 is NAVSP(Jsm,x)) =  2 2 m + n − 3,u ,u is 4n − 4,u u is 4n − 6,u ,u ,...,u is (s4 − 3s3 − 2s2 + 6s)x2s +2s−6 + (2s3 − 4s)x2s +2s−7+ 1 m 2 m−1 3 4 m−2  2n2 − 9n + 9 and v is n − 2 and sum of degree of these non-  3 2s2+s−7 2 2(s2−s) 1 (s − 3s)x + (s − s)x i f s,m is even adjacent vertices is 6m − 9,6m − 11,6m − 12,5m − 11,5m −   (s ≥ 4,m ≥ 4) i f s,m is odd 12,5m − 15,5m − 16 and 2m − 7 respectively. Combining all   these we get the required polynomial. (s ≥ 5,m ≥ 5)  4 2 2m2+4m−6 (s + 4s − 37)(m + m − 2)x +  2 2 (2s − 2)(m2 + m − 2)x2m +4m−7 + (s2 − s − 3)mx2m +3m−7 2 +(s2 − 2s + 1)x2m   i f s(s ≥ 5) is odd and i f m(m ≥ 4) is even  2 (s2 − 2s − 3)(m2 − m − 2)x2m −6   2 2m2−7 +(2s + 2)(m − m − 2)x  2 2 +(s2 + s − 3)mx2m −m−7 + (2s2 − 17)x2m −4m  i f s(s ≥ 4) is even and i f m (m ≥ 5) is odd

Proof. Let Jsm be a Jahangir graph with sm+1 vertices namely v1,v2,...,vsm,vsm+1. In a Jahangir graph Jsm, m vertices have degree 3, d(vsm+1) = m and the remaining vertices have de- Here NAVSP(U5,5,x) = gree 2. 8x23 + 7x21 + 7x20 + 77x16 + 14x15 + 12x12 + 6x11 + 3x5. Case (i) Both s (s ≥ 5) and m (m ≤ 5) are odd. The total number of non-adjacent vertices of degree 3 vertices Example 2.2. The non-adjacent vertex sum polynomial of the is s2 − 3s, those vertices adjacent to degree 3 vertices except 2 2 Umbrella u5,5,u6,7 and u7,6 are given in Figure 1, Figure 2 vsm+1 is 2s − 3s,vsm+1 is s − s and the remaining vertices in and Figure 3 respectively. the cycle is (s4 − 3s3 − 2s2 + 6s) and sum of degree of these

942 Non-adjacent vertex sum polynomial of Umbrella graphs, Jahangir graph, Tadpole graph and Lollipop graph — 943/944

2 2 2 non-adjacent vertices is 2s + s − 7, 2s + 2s − 7, 2(s − s) {u1v1} be the edge set of the graph Tn,k. Then Tn,k has n + k and 2s2 + 2s − 6 respectively. vertices and n + k edges. Case (ii) Both s (s ≥ 4) and m (m ≥ 4) are even. This proof Case (i) Both n (n ≥ 6) and k (k ≥ 4) are even. is similar to case (i). The total number of non-adjacent vertices of vk is (2n − 4), Case (iii) s(s ≥ 5) is odd and m(m ≥ 4) is even. vk−1 is (2n − 5),v2,v3,...,vk−2 and u3,u4,...,un−1 is (2n − The total number of non-adjacent vertices of degree 3 vertices 8)(2k − 1), u2,un and v1 is 3(2n − 5) and u1 is 2n − 6 and is (s2 − s − 3)m, those vertices adjacent to degree 3 vertices sum of degree of these non-adjacent vertices is 4k + 1,4k − 2 2 except vsm+1 is (2s − 2)(m + m − 2), vsm+1 is (s − 2s + 1), 1,4k − 2,4k − 3 and 4k − 5 respectively. and the remaining inner vertices in the cycle is (s2 + 4s − Case (ii) Both n(n ≥ 5) and k(k ≥ 3) are even. The result is 37)(m2 + m − 2) and sum of degree of these non-adjacent similar to Case (i). vertices is 2m2 +3m−7, 2m2 +4m−7, 2m2 and 2m2 +4m−6 Case (iii) n(n ≥ 5) is odd and k (k ≥ 4) is even. respectively. The total number of non-adjacent vertices of vk is 2n−3,vk−1 Case (iv) s (s ≥ 4) is even and m (m ≥ 5) is odd. is (2n−4),v2,v3,...,vk−2 and u3,u4,...,un−1 is (2n−7)(2k − The total number of non-adjacent vertices of degree 3 vertices 2),u2,un,v1 is 3(2n − 4) and u1 is 2n − 5 and sum of degree is m(s2 + s − 3), those vertices adjacent to degree 3 vertices of these non-adjacent vertices is 4k − 1,4k − 3,4k − 4,4k − 5 2 2 except vsm+1 is (2s + 2)(m − m − 2),vsm+1 is (2s − 17) and and 4k − 7 respectively. the remaining inner vertices in the cycle is (s2 −2s−3)(m2 − Case (iv) n (n ≥ 6) is even and k (k ≥ 3) is odd. The total m − 2) and sum of degree of these non-adjacent vertices is number of non-adjacent vertices of vk is (2n−5),vk−1 is 2n− 2 2 2 2 2m − m − 7, 2m − 7, 2m − 4m and 2m − 6 respectively. 6,v2,v3,...,vk−2 and u3,u4,...,un−1 is 2k(2n − 9), u2,un,v1 is 3(2n − 6) and u1 is 2n − 7 and sum of degree of these non-adjacent vertices is 4k + 3,4k + 1,4k,4k − 1 and 4k − 3 Put s = ,n = in the above theorem we have Example 2.4. 5 4 respectively. the graph. Example 2.6. Take n = 6,k = 4 in the above theorem we have the graph.

17 15 14 13 11 Here NAVSP(T6.4,x) = 8x + 7x + 28x + 21x + 6x 42 41 37 32 Here NAVSP(J5,4,x) = 144x + 144x + 68x + 16x . Theorem 2.7. Let Lm,n be a Lollipop graph with m + n ver- Theorem 2.5. Let Tn,k be a Tadpole graph with n + k vertices. Then the non-adjacent vertex sum polynomial of Lm,n is tices. Then the non-adjacent vertex sum polynomial of Tn,k is NAVSP(Lm,n,x) = NAVSP(Tn,k,x) = 2 2  (2m − 4)xn +5n−1 + (2m − 5)xn +5n−3 + (m − 5)(2n − 1) (2n − 4)x4k+1 + (2n − 5)x4k−1 + (2n − 8)(2k − 1)x4k−2   xn2+5n−4 + ( m − )xn2+4n−4 + (m − )nx2n−1 + (m − )x2n−3  4k−3 4k−5  2 5 1 3 +3(2n − 5)x + (2n − 6)x    i f m(m ≥ 7),n(n ≥ 5) are odd and i f n(n ≥ 6),k(k ≥ 4) are even    i f m(m ≥ 6),n(n ≥ 4) are even and i f n(n ≥ 5),k(k ≥ 3) are odd    n2+7n+3 n2+7n+1 n2+7n  4k−1 4k−3 4k−4 (2m − 5)x + (2m − 6)x + 2(m − 6)nx (2n − 3)x + (2n − 4)x + (2n − 7)(2k − 2)x  2 +(2m − 6)xn +6n−1 + (m − 1)nx2n−1 + (m − 4)x2n−3 +3(2n − 4)x4k−5 + (2n − 5)x4k−7    i f m(m ≥ 8) is even and i f n(n ≥ 5) is odd i f n(n ≥ 5 is odd and i f k(k ≥ 4) is even  2 2  (2m − 3)xn +3n−3 + (2m − 4)xn +3n−5 ( n − )x4k+3 + ( n − )x4k+1 + ( n − ) kx4k   2 5 2 6 2 9 2  2 2  +(m − 4)(2n − 2)xn +3n−6 + (2m − 4)xn +2n−5 +3(2n − 6)x4k−1 + (2n − 7)x4k−3    2n−1 2n−3  +(m − 1)nx + (m − 2)x i f n(n ≥ 6) is even and i f k(k ≥ 3) is odd  i f m(m ≥ 7) is odd and i f n(n ≥ 6) is even Proof. Let V = {{ui,v j/1 ≤ i ≤ n,1 ≤ j ≤ k} be the vertex set and E = {uiui+1,v jv j+1/1 ≤ i ≤ n − 1,1 ≤ j ≤ k − 1} ∪ Proof. Let V = {{ui,v j/1 ≤ i ≤ m,1 ≤ j ≤ n} be the vertex

943 Non-adjacent vertex sum polynomial of Umbrella graphs, Jahangir graph, Tadpole graph and Lollipop graph — 944/944

[6] set and E = {{uiui+1,v jv j+1/1 ≤ i ≤ m−1,1 ≤ j ≤ n−1}∪ A. Lourdusamy, S. Samuel Jeyaseelaan and T. Math- {u1v1} be the edge set of the graph Lm,n. Then Lm,n has m+n ivanan, On pebbling Jahangir graph, Gen. Math. Notes, m vertices and 2 + n edges. 5(2), 2011, 42-49. Case (i) Both m (m ≥ 6) and n(n ≥ 4) are even. [7] G. Sankari, S. Lavanya, Odd-Even Graceful Labeling Of The total number of non-adjacent vertices of vn is (2m − Umbrella And Tadpole Graphs, “International Journal of 4),vn−1 is 2m − 5,v2,v3,...,vn−2 is (m − 5)(2n − 1), v1 is Pure and Applied Mathematics”, 114 (6), 2017, 139-143. (2m − 5),u2,u3,...,um is n(m − 1) and u1 is (m − 3) and sum of degree of these non-adjacent vertices is n2 + 5n − 1, n2 + ????????? 5n−3, n2 +5n−4, n2 +4n−4, 2n−1 and 2n−3 respectively. Case (ii) Both m(m ≥ 7) and n(n ≥ 5) are odd. The result is ISSN(P):2319 − 3786 similar to case(i). Malaya Journal of Matematik Case (iii) m (m ≥ 8) is even and n (n ≥ 5) are odd. ISSN(O):2321 − 5666 ????????? The total number of non-adjacent vertices of vn is (2m − 5),vn−1 is 2m − 6,v2,v3,...,vn−2 is 2n(m − 6),v1 is 2m − 5,u3,u4,...,um is n(m − 1) and u1 is (m − 4) and sum of degree of these non-adjacent vertices is n2 + 7n + 3,n2 + 7n + 1,n2 + 7n,n2 + 6n − 1,2n − 1 and 2n − 3 respectively. Case (iv) m (m ≥ 7) is odd and n (n ≥ 6) is even. The total number of non-adjacent vertices of vn is (2m − 3),vn−1 is (2m − 4),v2,v3,...,vn−2 is (m − 4)(2n − 2),v1 is (2m−4),u2,u3,...,um is n(m−1) and u1 is m−2 and sum of degree of these non-adjacent vertices is n2 +3n−3,n2 +3n− 5,n2 +3n−6,n2 +2n−5,2n−1 and 2n−3 respectively.

Example 2.8. Take m=6,n=4 in the above theorem we have the graph.

Here 35 33 32 28 7 5 NAVSP(L6,4,x) = 8x + 7x + 7x + 7x + 20x + 3x

References [1] J. Devaraj, E.Sukumaran “On vertex polynomial”, Inter- national J. of Math.sci engg Appls, (IJMESA), 6(1), 2012, PP.371-380. [2] E.Ebin Raja Merly, A.M.Anto, “Vertex Polynomial for the Splitting graph of Comb and Crown, International journal of Emerging Technologies in Engineering Re- search(IJETER), 4(10),(2016), 53-55. [3] Frank Harary, 1872, “Graph Theory”, Addition – Wesly publishing company. [4] Hosoya.H and Harary.F “On the Matching properties of Three Fence Graphs”. Journal of Mathematical chemistry, 12, 1993, 211-218. [5] John P. Mcsoreley “Counting structures in the Mobius Ladder”, discrete mathematics, 184(1-3), 137-164.

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