Ratio Mathematica Volume 46, 2023 Superior eccentric domination polynomial Tejaskumar R* A Mohamed Ismayil† Abstract Superior distance involves the path which travels through the closed neighbourhood of both the vertices and the shortest path between them. This unique distance led to the advent of superior dominating sets and superior eccentric dominating sets. The former has a supe- rior neighbourin its compliment for every vertex in itself and the latter has a superior eccentric vertex in itself for every vertex in its compli- ment. The domination polynomials disuss the idea of total number of dominating sets and dominating sets of specific cardinality. This inspired us to conceptualise the idea of superior eccentric domination polynomial.In this paper, we introduce the superior eccentric dom- ination polynomial SED(G,φ) = ∑β l=γsed(G) |sed(G,l)|φ l where |sed(G,l)| is the number of all distinct superior eccentric dominating sets with cardinality l and γsed(G) is superior eccentric domination number. We find SED(G,) for family of wheel graphs and different standard graphs. The correlation between the coefficients of differ- ent SED polynomials are stated and proved. The motivation for this paper is to find a domination polynomial using distance concept in graphs. Eccentricity is a distance and eccentric dominating set was already existing. Keywords: Superior distance, superior eccentricity, superior eccen- tric domination polynomial. 2020 AMS subject classifications: 05C69, 11B83, 05C12. 1 *Jamal Mohamed College (Affiliated to Bharathidasan University), Tiruchirappalli, India. te- jaskumaarr@gmail.com. †Jamal Mohamed College (Affiliated to Bharathidasan University), Tiruchirappalli, India. amismayil1973@yahoo.co.in. 1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20, 2023. DOI: 10.23755/rm.v46i0.1082. ISSN: 1592-7415. eISSN: 2282-8214. ©Tejaskumar R et al. This paper is published under the CC-BY licence agreement. 257 Tejaskumar R and A Mohamed Ismayil 1 Introduction The shortest path between any two vertices is known as geodesic. The concept of distance in graphs always yields to cater the needs of applications in technology. There are many variants of distances in graphs. The shortest, longest, path involv- ing degree of vertices and chords. Kathiresan et al. [2007] introduced the superior distance in graphs. Let the path Dpq = N[p] ∪ N[q]. The shortest superior dis- tance between p to q is dD(p,q). Superior eccentricity of eD(p) = max{dD(q,p) : p,q ∈ V} Superior neighbour of p is dD(p) = min{dD(p,q) : q ∈ V −{p}}. A vertex p(q) is a superior neighbour of p if dD(p,q) = dD(p). The superior distance involves the shortest path between two vertices which travels through all their closed neighbourhoods. Using the superior distance the same authors Kathiresan and Marimuthu [2008] introduced the superior domination(SD-set).A set ⊆ V is called a SD-set if every vertex in S has superior neighbour in S-D. The SD-number is the cardinality of the minimum SD-set, denoted by γSed(G). The superior eccentric vertex p is given by dD (p,q) = eD(p). Here the adjacency between the vertices in S and its compli- ment is not mandatory. Bhanumathi and Abhirami [2017] introduced the superior eccentric domination(SED-set) in graphs. A set ⊆ V is an SED-set if every vertex in S-Dhas a superior eccentric vertex in S. The SED-number is the cardinality of the minimum SED-set, denoted by γSed(G). Along with being a superior domi- nating set if the same set has a superior eccentric vertex in itself for every vertex in the compliment of S then it becomes a superior eccentric dominating set. These two conditions play a vital role in the formation of a SED-set. Alikhani and Peng [2009] conceptualized the idea of domination polynomial, a domination polyno- mial consists of a coefficients which gives the number of dominating sets and the power of the variable denotes the cardinality of the dominating set which varies between one and the vertex cardinality of graph. They discussed and proved cer- tain properties which speaks of the corelation between the dominating sets. The motivation for this paper is to find a domination polynomial using distance concept in graphs. Eccentricity is a distance and eccentric dominating set was al- ready existing. Inspired by this work Ismayil and Tejaskumar [2020] introduced the eccentric domination polynomial. The eccentric dominating polynomial gives the idea about the number of eccentric dominating sets with different cardinal- ity and the symmetry in the coefficients of their polynomials werediscussed and proved. Superior distance, superior eccentric domination existed but there was a gap in the literature we did not have a formula which could find the total number of SED in a graph or a SED of specific cardinality, henceforth the same authors Ismayil and Tejaskumar extended the idea of domination polynomial to superior eccentric domination polynomial. In this paper, we discuss the concept of superior eccentric domination polynomial with an apt example, this concept was mainly in- 258 Superior eccentric domination polynomial troduced to find the total number of SED-sets of any graph. We found the formula which yields a SED polynomial for the family of wheel graphswhich helps us to easily find the total number of SED-sets of any cardinality at a given point of time for a wheel graph.We obtain the formulas and discuss the corelation between the coefficients of different SED polynomials. We tabulate the SED polynomials and their roots of different standard graphs. For all the undefined terminologies refer the book Graph Theory by Harary [2001]. 2 Superior eccentric domination polynomial Definition 2.1. The superior eccentric domination polynomial SED(G,φ) =∑β l=γsed(G) |sed(G,l)|φ l where |sed(G,l)| is the number of distinct superior ec- centric dominating sets (SED-sets) with cardinality l, β ∈ N and γsed(G) is supe- rior eccentric domination number. Example 2.1. . ℘4 ℘5℘3 ℘1 ℘2 Figure 1: Cricket graph Vertex Superior eccentricity Superior eccentric vertex eD(℘) ℘1 2 ℘2 ℘2 2 ℘1 ℘3 2 ℘5 ℘4 6 ℘1,℘2,℘3,℘5 ℘5 2 ℘3 Here we see the cricket graph has a SED-set {℘4} of cardinality 1, {℘1,℘4}, {℘2,℘4}, {℘3,℘4}, {℘4,℘5} SED sets of cardinality 2, {℘1,℘2,℘4},{℘1,℘3,℘4}, {℘1,℘4,℘5}, {℘2,℘3,℘4}, {℘2,℘4,℘5}, {℘3,℘4,℘5} SED sets of cardinality 3, {℘1,℘2,℘3,℘4}, {℘1,℘2,℘4,℘5},{℘1,℘3,℘4,℘5},{℘2,℘3,℘4,℘5} SED sets of cardinality 4 and {℘1,℘2,℘3,℘4,℘5} SED sets of cardinality 5. Therefore SED(G,φ) = φ 5 + 4φ 4 + 6φ 3 + 4φ 2 + φ. 259 Tejaskumar R and A Mohamed Ismayil 3 Superior eccentric domination polynomial of wheel graph Definition 3.1. Superior eccentric domination polynomial of a wheel graph Wβ is given by SED(Wβ,φ) = ∑β l=γsed(Wβ) |sed(Wβ, l)|φ,l where |sed(Wβ, l)| is the number of distinct SED-sets with cardinality l and γsed(Wβ) is SED-number of wheel. Observation 3.1. . 1. SED(Wβ,φ) = (1 + φ)β −1, for β = 4,5. 2. SED(Wβ,φ) = (1 + φ)β, for β = 6. Theorem 3.1. For a wheel graph Wβ of order β, 1. |sed(Wβ, l)| = |sed(Wβ−1, l−1)|+|sed(Wβ−1, l)| where l ≤ β and β ≥ 7. 2. SED(Wβ,φ) = φ SED(Wβ−1,φ) + SED(Wβ−1,φ). 3. SED(Sβ,φ) = φ(φ + 1)β−1, for all β ≥ 7. Proof: Let V (Wβ) = {℘1,℘2, . . .℘β}. 1. Since |sed(Wβ, l)| =β−1 Cl−1, |sed(Wβ, l−1)| =β−2 Cl−2 and |sed(Wβ−1, l)| =β−2 Cl−1. But β−1Cl−1 =β−2 Cl−2 +β−2 Cl−1. Therefore |sed(Wβ, l)| = |sed(Wβ−1, l−1)|+ |sed(Wβ−1, l)|. 2. By theorem-wheelTHM01 − (1) we have |sed(Wβ, l)| = |sed(Wβ−1, l − 1)|+ |sed(Wβ−1, l)|. When l = 1, |sed(Wβ,1)| = |sed(Wβ−1,0)|+ |sed(Wβ−1,1)| =⇒ φ, |sed(Wβ,1)| = φ, |sed(Wβ−1,0)|+ φ, |sed(Wβ−1,1)|. When l = 2, |sed(Wβ,2)| = |sed(Wβ−1,1)|+ |sed(Wβ−1,2)| =⇒ φ,2 |sed(Wβ,2)| = φ,2 |sed(Wβ−1,1)|+ φ,2 |sed(Wβ−1,2)|. When l = 3, |sed(Wβ,3)| = |sed(Wβ−1,2)|+ |sed(Wβ−1,3)| =⇒ φ 3|sed(Wβ,3)| = φ 3|sed(Wβ−2,1)|+ φ 3|sed(Wβ−1,3)|. 260 Superior eccentric domination polynomial When l = 4, |sed(Wβ,4)| = |sed(Wβ−1,3)|+ |sed(Wβ−1,4)| =⇒ φ 4|sed(Wβ,4)| = φ 4|sed(Wβ−1,3)|+ φ 4|sed(Wβ−1,4)|. ... When l = β −1, |sed(Wβ,β −1)| = |sed(Wβ−1,β −2)| +|sed(Wβ−1,β −1)| =⇒ φ β −1|sed(Wβ,β −1)| = φ β −1|sed(Wβ−1,β −2) +φ β −1|sed(Wβ−1,β −1)|. When l = β, |sed(Wβ,β)| = |sed(Wβ−1,β −1)|+ |sed(Wβ−1,β)| =⇒ φ β|sed(Wβ,β)| = φ β|sed(Wβ−1,β −1)|+ φ β|sed(Wβ−1,β)|. Therefore φ |sed(Wβ,1)|+φ 2|sed(Wβ,2)|+φ 3|sed(Wβ,3)|+φ 4|sed(Wβ,4)|+ · · ·+ φ β−1|sed(Wβ,β −1)|+ φ β|sed(Wβ,β)| = φ |sed(Wβ−1,0)|+φ |sed(Wβ−1,1)|+φ 2|sed(Wβ−1,1)|+φ 2|sed(Wβ−1,2)|+ φ 3|sed(Wβ−2,1)|+ φ 3|sed(Wβ−1,3)|+ φ 4|sed(Wβ−1,3)|+ φ 4|sed(Wβ−1,4)| + · · ·+ φ β−1|sed(Wβ−1,β −2)| + φ β−1|sed(Wβ−1,β −1)|+ φ β|sed(Wβ−1,β −1)|+ φ β|sed(Wβ−1,β)|. = φ |sed(Wβ−1,0)|+ φ 2|sed(Wβ−1,1)|+ φ 3|sed(Wβ−2,1)|+ φ 4|sed(Wβ−1,3)| + · · ·+ φ β−1|sed(Wβ−1,β −2)|+ φ β|sed(Wβ−1,β −1)|+ φ |sed(Wβ−1,1)| + φ 2|sed(Wβ−1,2)|+ φ 3|sed(Wβ−1,3)|+ φ 4|sed(Wβ−1,4)|+ . . . + φ β−1|sed(Wβ−1,β −1)|+ φ β|sed(Wβ−1,β)| = φ [φ |sed(Wβ−1,1)|+ φ 2|sed(Wβ−1,2)|+ φ 3|sed(Wβ−1,3)| + φ 4|sed(Wβ−1,4)|+ · · ·+ φ β−1|sed(Wβ−1,β −1)|] + φ |sed(Wβ−1,1)| + φ 2|sed(Wβ−1,2)|+ φ 3|sed(Wβ−1,3)|+ φ 4|sed(Wβ−1,4)| + · · ·+ φ β−1|sed(Wβ−1,β −1)| Since |sed(Wβ−1,0)| = |sed(Wβ−1,β)| = 0. = φ ∑β−1 l=1 |sed(Wβ−1, l)|φ l + ∑β−1 l=1 |sed(Wβ−1, l)|φ l. SED(Wβ,φ) = φ SED(Wβ−1,φ) + SED(Wβ−1,φ). 1. By mathematical induction (MI). 261 Tejaskumar R and A Mohamed Ismayil It is true for β = 7. SED(Wβ−1,φ) = φ(φ + 1) 7−1 = φ(φ + 1)6 = φ(φ + 1)3(φ + 1)3 = φ 7 + 6φ 6 + 15φ 5 + 20φ 4 + 15φ 3 + 6φ 2 + 1. Assume it is true ∀ N less than ′β′. SED(Wβ,φ) = φ(1 + φ) (β−1)−1 = φ(1 + φ)β−2 For′β′, SED(Wβ,φ) = φ SED(Wβ−1,φ) + SED(Wβ−1,φ) usingtheorem−3.1− (2) = φ [φ(φ + 1)β−2] + φ(φ + 1)β−2 = φ(φ + 1)β−1 ∴ Proved ∀ ′β′. Table: |sed(Wβ, l)| is the number of superior eccentric dominating sets of Wβ with cardinality l where 1 ≤ l ≤ 15. H H H H HHβ l 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0 2 0 0 3 0 0 0 4 4 6 4 1 5 5 10 10 5 1 6 1 5 10 10 5 1 7 1 6 15 20 15 6 1 8 1 7 21 35 35 21 7 1 9 1 8 28 56 70 56 28 8 1 10 1 9 36 84 126 126 84 36 9 1 11 1 10 45 120 210 252 210 120 45 10 1 12 1 11 55 165 330 462 462 330 165 55 11 1 13 1 12 66 220 495 792 924 792 495 220 66 12 1 14 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 15 1 14 91 364 1001 2002 3003 3423 3003 2002 1001 364 91 14 1 Theorem 3.2. The following properties for the co-efficients of SED(Wβ,φ) holds. 1. |sed(Wβ,1)| = 1 for all β ≥ 6. 2. |sed(Wβ,β)| = 1, ∀ β ≥ 4. 3. |sed(Wβ,β −1)| = β −1, ∀ β ≥ 6. 262 Superior eccentric domination polynomial 4. |sed(Wβ,β −2)| = (β−1)(β−2) 2 , ∀ β ≥ 6. 5. |sed(Wβ,β −3)| = (β−1)(β−2)(β−3) 6 , ∀ β ≥ 6. 6. |sed(Wβ,β −4)| = (β−1)(β−2)(β−3)(β−4) 24 , ∀ β ≥ 6. 7. |sed(Wβ, l)| = |sed(Wβ,β − l + 1)|, ∀ β ≥ 6. 8. If SEDβ = ∑β l=1 |sed(Wβ, l)|, ∀ β ≥ 6, then SEDβ = 2(SEDβ−1), ∀ β ≥ 7. 9. SEDβ =Total number of SED-sets in Wβ = 2β−1, ∀ β ≥ 6. Proof: 1. Let V (Wβ) = {℘1,℘2,℘3, . . .℘β}. In a wheel graph Wβ all the vertices form a superior neighbour of central vertex ℘1 except itself. Therefore the only set with single cardinality D = {℘1} forms the superior eccentric dominating set of wheel graph Wβ where β ≥ 6. 2. The vertex set V (Wβ) forms the superior eccentric dominating set |sed(Wβ,β)| = 1 for all β ≥ 4. 3. By MI on ′β′. For β = 6, |sed(W6,6−1)| = |sed(W6,5)| = 5. Assume it is true ∀ N less than ′β′. For ′β′, By theorem-3.1-(2) and 3.2-(2) |sed(Wβ,β −1)| = |sed(Wβ−1,β −2)|+ |sed(Wβ−1,β −1)| = (β −2) + 1 = β −1 ∴ Proved ∀ ′β′. 4. By MI on ′β′. For β = 6, |sed(W6,6−2)| = |sed(W6,4)| = 10. For β = 7, |sed(W7,7−2)| = |sed(W7,5)| = 15. Assume it is true ∀ N less than ′β′. 263 Tejaskumar R and A Mohamed Ismayil For ′β′, By theorem-3.1 and 3.2-(3) |sed(Wβ,β −2)| = |sed(Wβ−1,β −3)|+ |sed(Wβ−1,β −2)| = (β −2)(β −3) 2 + (β −2) = (β −2)(β −3) + 2(β −2) 2 = (β −2)(β −3 + 2) 2 = (β −2)(β −1) 2 ∴ Proved ∀ ′β′. 5. By MI on ′β′. For β = 6, |sed(W6,6−3)| = |sed(W6,3)| = 10. For β = 7, |sed(W7,7−3)| = |sed(W7,4)| = 20. Assume it is true ∀ N less than ′β′. For ′β′, By theorem-3.1 and 3.2-(4) |sed(Wβ,β −3)| = |sed(Wβ−1,β −4)|+ |sed(Wβ−1,β −3)| = (β −2)(β −3)(β −4) 6 + (β −2)(β −3) 2 = (β −2)(β −3)(β −4 + 3) 6 = (β −1)(β −2)(β −3) 6 ∴ Proved ∀ ′β′. 6. By MI on ′β′. The result is true for β = 6, |sed(W6,6−4)| = |sed(W6,2)| = 5. For β = 7, |sed(W7,7−4)| = |sed(W7,3)| = 15. Assume it is true ∀, N < β. 264 Superior eccentric domination polynomial For ′β′, By theorem-3.1 and 3.2-(5) |sed(Wβ,n−4)| = |sed(Wβ−1,β −5)|+ |sed(Wβ−1,β −4)| = (β −2)(β −3)(β −4)(β −5) 24 + (β −2)(β −3)(β −4) 6 = (β −2)(β −3)(β −4)(β −5 + 4) 24 = (β −1)(β −2)(β −3)(β −4) 24 ∴ Proved ∀ ′β′. 7. By MI on ′β′. The result is true for β = 6. |sed(W6,2)| = |sed(W6,6−2 + 1)| = |sed(W6,5)| = 5 |sed(W7,3)| = |sed(W7,7−3 + 1)| = |sed(W7,4)| = 20. Assume it is true ∀ N less than ′β′. For ′β′, by theorem-3.1 we have |sed(Wβ, l)| = |sed(Wβ−l, l−1)|+ |sed(Wβ−1, l)| = |sed(Wβ−1,(β −1− (l−1) + 1))| + |sed(Wβ−1,(β −1− (l) + 1))| = |sed(Wβ−1,(β −1− l + 1 + 1))| + |sed(Wβ−1,(β −1− l + 1))| = |sed(Wβ−1,(β − l + 1))| + |sed(Wβ−1,(β − l))| = |sed(Wβ,(β − l + 1))| ∴ Proved ∀ ′β′. 8. SEDβ = ∑β l=1 |sed(Wβ, l)| By theorem-3.1 we have SEDβ = β∑ l=1 [|sed(Wβ−1, l−1)|+ |sed(Wβ−1, l)|] = β−1∑ l=1 |sed(Wβ−1, l)|+ β−1∑ l=1 |sed(Wβ−1, l)| = SEDβ−1 + SEDβ−1 SEDβ = 2(SEDβ−1) 265 Tejaskumar R and A Mohamed Ismayil 9. By MI on ′β′. When β = 6, SED6 = 2 6−1 = 25 = 32. SED7 = 2 7−1 = 26 = 64. Assume it is true ∀ N less than ′β′. SEDβ−1 = 2 β−1−1 = 2β−2 For ′β′, SEDβ = 2[SEDβ−1] from theorem −3.2− (8) = 2[2β−2] = 2β−1 ∴ Proved ∀ ′β′. Hence the theorem. Remark 3.1. . 1. For any graph G 0 is one of the root of every SED(G,φ). 2. A graph with more than 3 pendant vertices has at least 2 real roots. SED(G,φ) of different standard graphs and their roots are tabulated be- low: Graph Figure Superior eccentric domination polynomial SED(G,φ) Roots Diamond graph ℘1 ℘4 ℘2 ℘3 φ 4 + 4φ 3 + 5φ 2 φ 1 = 0, φ 2 = −0.4563, φ 3 = −1.7718 + 1.1151i, φ 4 = −1.7718−1.1151i. Tetrahedral graph ℘2 ℘1 ℘3 ℘4 φ 4 + 4φ 3 + 6φ 2 + 4φ φ 1 = 0, φ 2 = −2, φ 3 = −1 + i, φ 4 = −1− i. 266 Superior eccentric domination polynomial Graph Figure Superior eccentric domination polynomial SED(G,φ) Roots Paw graph ℘2 ℘3 ℘1 ℘4 φ 4 + 3φ 3 + 3φ 2 + φ φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −1. Banner graph ℘3 ℘4 ℘1 ℘2 ℘5 φ 5 + 4φ 4 + 5φ 3 + 3φ 2 φ 1 = 0, φ 2 = −2.4656, φ 3 = −0.7672 + 0.7926i, φ 4 = −0.7672−0.7926i. Graph Figure Superior eccentric domination polynomial SED(G,φ) Roots Fork graph ℘2 ℘3 ℘1 ℘4 ℘5 φ 5 + 4φ 4 + 5φ 3 + 2φ 2 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −2. (3,2)-Tadpole graph ℘2 ℘3 ℘4 ℘1 ℘5 φ 5 + 4φ 4 + 3φ 3 + φ 2 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −2. Kite graph ℘3 ℘4 ℘1 ℘5 ℘2 φ 5 + 5φ 4 + 6φ 3 + 2φ 2 φ 1 = 0, φ 2 = −0.378 +0.1877i, φ 3 = −0.378 −0.1877i, φ 4 = −2.122 +1.0538i, φ 5 = −2.122 −1.053i. 267 Tejaskumar R and A Mohamed Ismayil Graph Figure Superior eccentric domination polynomial SED(G,φ) Roots House graph ℘2 ℘3 ℘1 ℘4 ℘5 φ 5 + 5φ 4 + 6φ 3 φ 1 = 0, φ 2 = −0.3076 +0.3182i, φ 3 = −0.3076 −0.3182i, φ 4 = −2.1924 +0.5479i, φ 5 = −2.1924 −0.5479i. House X graph ℘2 ℘3 ℘1 ℘4 ℘5 φ 5 + 5φ 4 + 6φ 3 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −0.382, φ 5 = −2.618. Dart graph ℘3 ℘4 ℘1 ℘5 ℘2 φ 5 + 4φ 4 + 6φ 3 + 3φ 2 φ 1 = 0, φ 2 = −1, φ 3 = −32 + √ 3 2 i, φ 4 = −32 − √ 3 2 i. Johnson solid skeleton 12 graph ℘2 ℘1 ℘3 ℘4 ℘5 φ 5 + 5φ 4 + 10φ 3 + 10φ 2 + 5φ φ 1 = 0, φ 2 = −0.691 +0.9511i, φ 3 = −0.691 −0.9511i, φ 4 = −1.809 +0.5878i, φ 5 = −1.809 −0.5878i. Net graph ℘5 ℘6 ℘3 ℘4 ℘1 ℘1 φ 6 + 3φ 5 + 3φ 4 + φ 3 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −1. A graph ℘3 ℘4 ℘1 ℘5 ℘2 ℘6 φ 6 + 4φ 5 + 6φ 4 + 4φ 3 + φ 2 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −1, φ 5 = −1. 268 Superior eccentric domination polynomial Graph Figure Superior eccentric domination polynomial SED(G,φ) Roots 4-polynomial graph 1 ℘2 ℘3℘1 ℘5℘4 ℘6 φ 6 + 4φ 5 + 4φ 4 φ 1 = 0, φ 2 = −2. Antenna graph ℘2 ℘1 ℘3 ℘4 ℘5 ℘6 φ 6 + 4φ 5 + 3φ 4 + φ 3 φ 1 = 0, φ 2 = −1, φ 3 = −2, φ 4 = −1. Octahedral graph ℘4 ℘3℘2 ℘1 ℘5 ℘6 φ 6 + 6φ 5 + 15φ 4 + 20φ 3 +15φ 2 + 6φ φ 1 = 0, φ 2 = −2, φ 3 = −0.5 +0.866i, φ 4 = −0.5 −0.866i, φ 5 = −1.5 +0.866i, φ 6 = −1.5 −0.866i. Cubical graph ℘3 ℘4 ℘5 ℘6 ℘1 ℘2 ℘8℘7 φ 8 + 8φ 7 + 28φ 6 + 56φ 5 +68φ 4 + 48φ 3 + 16φ 2 φ 1 = 0, φ 2 = −0.6714 +0.5756i, φ 3 = −0.6714 −0.5756i, φ 4 = −0.8352 +1.4854i, φ 5 = −0.8352 −1.4854i, φ 6 = −2.4934 +0.9097i, φ 7 = −2.4934 −0.9097i. Wagner graph ℘1 ℘8 ℘4 ℘5 ℘2 ℘7 ℘3 ℘6 φ 8 + 8φ 7 + 24φ 6 +32φ 5 + 16φ 4 φ 1 = 0, φ 2 = −2, φ 3 = −2, φ 4 = −2, φ 5 = −2. 269 Tejaskumar R and A Mohamed Ismayil 4 Conclusions In this paper SED polynomial for a graph was defined. Formula to find the SED polynomials of family of wheel graphs were stated and proved. Corelation between the coefficients of SED polynomials were discussed. The SED polyno- mial and its roots for different standard graphs are tabulated. In the course of future work the graphs can be classified based on the roots of SED polynomials. The similarproperties of coefficients based on similar roots for different standard graphs can be discussed. The same concept can be extended to other domination parameter. References S. Alikhani and Y.-h. Peng. Introduction to domination polynomial of a graph. arXiv preprint arXiv:0905.2251, 2009. M. Bhanumathi and R. M. Abirami. Superior eccentric domination in graphs. 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