EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1449-1462 ISSN 1307-5543 – ejpam.com Published by New York Business Global Polynomial Representations and Degree Sequences of Graphs Resulting From Some Graph Operations Jayhan Cruz1,∗, Gina Malacas1,2, Sergio R. Canoy, Jr.1,2 1 Department of Mathematics and Statistics, College of Science and Mathematics MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Center for Mathematical and Theoretical Physical Sciences-PRISM MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G = (V (G), E(G)) be a graph with degree sequence ⟨d1, d2, · · · , dn⟩, where d1 ≥ d2 ≥ · · · ≥ dn. The polynomial representation of G is given by fG(x) = n∑ i=1 xdi = ∆(G)∑ k=1 akx k, where ak is the number of vertices of G having degree k for each i = 1, 2, · · ·n = ∆(G). In this paper, we give the polynomial representation of the complement and line graph of a graph, the shadow graph, complementary prism, edge corona, strong product, symmetric product, and disjunction of two graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: polynomial representation, line graph, edge corona, shadow graph, complementary prism, strong product, symmetric product, disjunction 1. Introduction Let G = (V (G), E(G)) be a graph on n vertices and let ∆(G) be the maximum degree of G. If s1, s2, · · · , sn are the degrees of the vertices of G, where s1 ≥ s2 ≥ · · · ≥ sn, then the sequence ⟨s1, s2, · · · , sn⟩ is called the degree sequence of G. Here, s1 = ∆(G). The polynomial fG(x) = n∑ i=1 xsi is called the polynomial representation of G. A degree sequence ⟨s1, s2, . . . , sn⟩ of nonnegative integers is said to be graphic if a simple graph G with degree sequence ⟨s1, s2, . . . , sn⟩ can be found (see [1]). Using the polynomial representation of a graph, we can alternatively define a polynomial P (x) to be graphic if there exists a graph G such that P (x) = fG(x). It is easy to verify not every polynomial is graphic. The degree sequence of a graph had been investigated in [2], [3], [4], [5], [6], ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5274 Email addresses: jayhan.cruz@g.msuiit.edu.ph (J. Cruz), gina.malacas@g.msuiit.edu.ph (G. Malacas), sergio.canoy@g.msuiit.edu.ph (S. Canoy, Jr.) https://www.ejpam.com 1449 © 2024 EJPAM All rights reserved. Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1450 [9], and [12]). Erdős and Gallai in [4] obtained a necessary and sufficient condition for a given polynomial to be graphic. The polynomial representations and degree sequences of the the join, corona, lexicographic product, Cartesian product, and Tensor product of two graphs had been obtained by Canoy et al. in [8]. These graphs were also investigated for other graph parameters in previous studies (see [7], [10], and [11]). In this present study, the authors endeavored to determine expressions for the poly- nomial representations of the complement and line graph of a graph, shadow graph, com- plementary prism, edge corona, strong product, symmetric difference, and disjunction of two graphs. 2. Terminologies and Notations Let G = (V (G), E(G)) be a simple undirected graph. The distance between two vertices u and v of G, denoted by dG(u, v), is equal to the length of a shortest path connecting u and v. Any path connecting u and v of length dG(u, v) is called a u-v geodesic. The open neighborhood of a vertex v of G is the set NG(v) = {u ∈ V (G) : uv ∈ E(G)} and its closed neighborhood is the set NG[v] = NG(v) ∪ {v}. The open neighborhood of a subset S of V (G) is the set NG(S) = ∪v∈SNG(v) and its closed neighborhood is the set NG[S] = NG(S) ∪ S. The degree of v, denoted by degG(v), is equal to |NG(v)|. The maximum degree of G, denoted by ∆(G), is equal to max{degG(v) : v ∈ V (G)}. Suppose ∆(G) = n. For each i = 1, 2, · · ·n, let ai be the number of vertices of G with degree i ≥ 0. Then the polynomial fG(x) = n∑ i=1 aix i is called the polynomial representation of G. Equivalently, fG(x) = ∑ v∈V (G) x|NG(v)|. Let G and H be graphs. The complement of G, denoted by G is the graph with V (G) = V (G) and vw ∈ E(G) if and only if vw /∈ E(G). The line graph L(G) of G is the graph with V (L(G)) = E(G) and e1e2 ∈ E((L(G)) if and only if e1 and e2 have a common vertex in G. The shadow graph D2(G) of G is the graph obtained by taking two copies of G, say G1 and G2, and joining each vertex u ∈ V (G1) to the neighbors of the corresponding vertex u′ ∈ V (G2). The complementary prism GG is the graph formed from the disjoint union of G and its complement G by adding a perfect matching between corresponding vertices of G and G. For each v ∈ V (G), let v denote the vertex in G corresponding to v. In simple terms, the graph GG is formed from G ∪ G by adding the edge vv for every vertex v ∈ V (G). The edge corona G ⋄ H of graphs G and H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each of the end vertices u and v of every edge uv in G to every vertex of the copy Huv of H (that is forming the join ⟨{u, v}⟩+Huv for each uv ∈ E(G)). The strong product G⊠H of graphs G and H is the graph with vertex set V (G)× V (H) and (u, v) is adjacent with (u′, v′) whenever [uu′ ∈ E(G) and v = v′] or [vv′ ∈ E(H) and u = u′] or [uu′ ∈ E(G) and vv′ ∈ E(H)]. The symmetric difference G⊕H of graphs G and H is the graph with vertex set V (G)×V (H) and (u, v) is adjacent with (u′, v′) whenever [uu′ ∈ E(G)] or [vv′ ∈ E(H)] but not both. Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1451 The disjunction G∨H of graphs G and H is the graph with vertex set V (G)×V (H) and (u, v) is adjacent with (u ′ , v ′ ) whenever uu ′ ∈ E(G) or vv ′ ∈ E(H). 3. Results The first result gives the polynomial representation of the complement of a graph. Theorem 1. Let G be a non-trivial graph of order n. Then fG(x) = xn−1fG( 1 x). Proof. Let v ∈ V (G). Then NG(v) = {w ∈ V (G) : w ∈ V (G) \ NG[v]}. This implies that |NG(v)| = n− |NG(v)| − 1. It follows that fG(x) = ∑ v∈V (G) x|NG(v)| = ∑ v∈V (G) xn−|NG(v)|−1 = xn−1 ∑ v∈V (G) x−|NG(v)| = xn−1fG( 1 x ). Theorem 2. Let G be a non-trivial graph of order n. If the degree sequence of G is ⟨d1, d2, · · · dn⟩, then the degree sequence of G is ⟨n− dn − 1, n− dn−1 − 1, · · · , n− d2 − 1, n− d1 − 1⟩. Proof. By Theorem 1, fG(x) = n∑ j=1 x|NG(v)| = n∑ j=1 xn−dj−1. Hence, the degree sequence of G is ⟨n− dn − 1, n− dn−1 − 1, · · · , n− d1 − 1⟩. This proves the assertion. Next, we give the polynomial representation of the line graph of a graph. Theorem 3. Let G be a non-trivial connected graph. Then fL(G)(x) = 1 x2 ∑ uv∈E(G) x|NG(u)|+|NG(v)|. Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1452 Proof. Let e = uv ∈ V (L(G)). Then |NL(G)(e)| = |NG(u)| + |NG(v)| − 2. It follows that fL(G)(x) = ∑ e∈V (L(G)) x|NL(G)(e)| = ∑ uv∈V (L(G)) x|NG(u)|+|NG(v)|−2 = 1 x2 ∑ uv∈E(G) x|NG(u)|+|NG(v)|. Corollary 1. Let G be a non-trivial r-regular connected graph of size p. Then fL(G)(x) = px2r−2. Proof. Since G is r-regular, |NG(v)| = r for all v ∈ V (G). By Theorem 3, we have fL(G)(x) = 1 x2 ∑ uv∈E(G) x|NG(u)|+|NG(v)| = 1 x2 ∑ uv∈E(G) x2r = px2r−2. Theorem 4. Let G be a connected graph of size p and let H any graph of order n. Then fG⋄H(x) = fG(x n+1) + px2fH(x). Proof. Let v ∈ V (G ⋄H). If v ∈ V (G), then NG⋄H(v) = NG(v) ∪ [∪u∈NG(v)V (Huv)]. If v ∈ V (He) for e = uw ∈ E(G), then NG⋄H(v) = NHe(v) ∪ {u,w}. Thus, fG⋄H(x) = ∑ v∈V (G⋄H) x|NG⋄H(v)| = ∑ v∈V (G) x|NG⋄H(v)| + ∑ v∈V (G⋄H)\V (G) x|NG⋄H(v)| = ∑ v∈V (G) x|NG(v)|+n|NG(v)| + ∑ e∈E(G) ∑ v∈V (He) x|NHe (v)|+2 = ∑ v∈V (G) x(n+1)|NG(v)| + px2 ∑ v∈V (H) x|NH(v)| = fG(x n+1) + px2fH(x). Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1453 The following result is immediate from the above result: Corollary 2. Let G be a connected graph of size p and let H be an r-regular graph of order n. Then fG⋄H(x) = fG(x n+1) + pnxr+2. In particular, fG⋄Kn(x) = fG(x n+1) + pnxn+1. Corollary 3. Let m and n be positive integers. Then (i) fPm⋄Pn(x) = (m− 2)x2n+2 + 2xn+1 + (m− 1)[(n− 2)x4 + 2x3] for m,n ≥ 2; (ii) fPm⋄Cn(x) = (m− 2)x2n+2 + 2xn+1 + (m− 1)nx4 for m ≥ 2 and n ≥ 3; and (iii) fCm⋄Cn(x) = mx2n+2 +mnx4 for m,n ≥ 3. Proof. Clearly, |E(Pr)| = r− 1, |E(Cs)| = s, fPr(x) = 2x+(r− 2)x2 and fCs(x) = sx2 for positive integers r ≥ 2 and s ≥ 3. By Theorem 4, we find that (i), (ii), and (iii) hold. Theorem 5. Let G be a connected graph of size p and let H be any graph with degree sequences ⟨d1, d2, · · · dm⟩ and ⟨r1, r2, · · · rn⟩, respectively. Then the terms of the degree sequence of G⋄H are the elements of the set {(n+1)di : 1 ≤ i ≤ m}∪{rj+2 : 1 ≤ j ≤ n}, where p consecutive terms of the degree sequence are rj + 2 for each j with 1 ≤ i ≤ n. Proof. The polynomial representations of G and H are, respectively, fG(x) = m∑ i=1 xdi and fH(x) = n∑ j=1 xrj . By Theorem 4, fG⋄H(x) = m∑ i=1 x(n+1)di + px2 n∑ j=1 xrj = m∑ i=1 x(n+1)di + p n∑ j=1 xrj+2. It follows that the terms of the degree sequence of G ⋄ H are the elements of the set {(n + 1)di : 1 ≤ i ≤ m} ∪ {rj + 2 : 1 ≤ j ≤ n}. Moreover, p consecutive terms of the degree sequence are rj + 2 for each j with 1 ≤ j ≤ n. Theorem 6. Let G be a non-trivial connected graph and let G1 and G2 be copies of G in the shadow graph D2(G). Then fD2(G)(x) = 2fG(x 2). Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1454 Proof. Let v ∈ V (G1) and let v′ be the vertex of G2 corresponding to v. Then ND2(G)(v) = NG1(v) ∪NG2(v ′) = ND2(G)(v ′). This implies that |ND2(G)(v)| = |ND2(G)(v ′)| = 2|NG(v)|. Thus, fD2(G)(x) = ∑ v∈V (D2(G)) x|ND2(G)(v)| = ∑ v∈V (G1) x|ND2(G)(v)| + ∑ v′∈V (G2) x|ND2(G)(v ′)| = 2 ∑ v∈V (G) x2|NG(v)| = 2fG(x 2). Corollary 4. Let m and n be positive integers such that m ≥ 2 and n ≥ 3. Then (i) fD2(Pm)(x) = 2(m− 2)x4 + 4x2; (ii) fD2(Cn)(x) = 2nx4; and (iii) fD2(Sm)(x) = 2x2m + 2mx2, where Sm is a star of order m+ 1. Proof. The polynomial representations of Pm, Cn, and Sm are, respectively, fPm(x) = (m− 2)x2 + 2x, fCn(x) = nx2, and fSm(x) = xm +mx. It follows from Theorem 6 that fD2(Pm)(x) = 2fPm(x 2) = 2(m− 2)x4 + 4x2, fD2(Cn)(x) = 2fCn(x 2) = 2nx4, and fD2(Sm)(x) = 2fSm(x 2) = 2x2m + 2mx2. This proves the assertion. Theorem 7. Let G be a non-trivial connected graph. Then a ∈ R is a zero of fD2(G)(x) if and only if a2 is a zero of fG(x). Proof. By Theorem 6, fD2(G)(x) = 2fG(x 2). Hence, if a is a zero of fD2(G)(x), then 2fG(a 2) = 0. This implies that a2 is a zero of fG(x). Conversely, if a2 is a zero of fG(x), then fD2(G)(a) = 2fG(a 2) = 0. Thus, a is a zero of fD2(G)(x). Theorem 8. Let G be a non-trivial connected graph with degree sequence ⟨d1, d2, · · · dn⟩. Then the degree sequence of D2(G) is ⟨2d1, 2d1, 2d2, 2d2, · · · , 2dn, 2dn⟩. Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1455 Proof. Given the degree sequence ⟨d1, d2, · · · dn⟩ of G, it follows that fG(x) = ∑n i=1 x di . Hence, by Theorem 6, fD2(G)(x) = 2fG(x 2) = 2 n∑ i=1 x2di . It follows that the degree sequence of D2(G) is ⟨2d1, 2d1, 2d2, 2d2, · · · , 2dn, 2dn⟩. Theorem 9. Let G be a non-trivial connected graph of order n. Then fGG(x) = xfG(x) + xnfG( 1 x ). Proof. Let v ∈ V (G). Then NGG(v) = NG(v) ∪ {v} and NGG(v) = NG(v) ∪ {v} = {z ∈ V (G) : z ∈ V (G) \NG[v]} ∪ {v}. Thus, |NGG(v)| = |NG(v)|+1 and |NGG(v)| = (n−|NG[v]|)+1 = n−|NG(v)|. Therefore, fGG(x) = ∑ p∈V (GG) x|NGG(p)| = ∑ v∈V (G) x|NGG(v)| + ∑ v∈V (G) x|NGG(v)| = ∑ v∈V (G) x|NG(v)|+1 + ∑ v∈V (G) xn−|NG(v)| = x ∑ v∈V (G) x|NG(v)| + xn ∑ v∈V (G) x−|NG(v)| = xfG(x) + xnfG( 1 x ). Corollary 5. Let n be a positive integer. Then (i) fPnPn (x) = 2xn−1 + (n− 2)xn−2 + (n− 2)x3 + 2x2 for n ≥ 2; (ii) fCnCn (x) = nxn−2 + nx3 for n ≥ 3; and (iii) fSnSn (x) = xn+1 + nxn + nx2 + x for n ≥ 2. Proof. From Theorem 9 and the polynomial representations fPn(x) = (n− 2)x2 + 2x, fCn(x) = nx2, and fSn(x) = xn + nx of Pn, Cn, and Sn, respectively, we have fPnPn (x) = x[(n−2)x2+2x]+xn[(n−2) 1 x2 +2 1 x ] = 2xn−1+(n−2)xn−2+(n−2)x3+2x2, Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1456 fCnCn (x) = x(nx2) + xn(n 1 x2 ) = nxn−2 + nx3, and fSnSn (x) = x(xn + nx) + xn+1( 1 xn + n 1 x ) = xn+1 + nxn + nx2 + x. This proves the assertion. Theorem 10. Let G be a non-trivial connected graph with degree sequence ⟨d1, d2, · · · dn⟩. Then the terms of the degree sequence of GG are the elements of the set {di + 1 : 1 ≤ i ≤ n} ∪ {n− di : 1 ≤ i ≤ n}. Proof. From the polynomial representation fG(x) = p∑ i=1 xdi of G and from Theorem 9, we find that fGG(x) = xfG(x) + xnfG 1 x ) = x n∑ i=1 xdi + xn n∑ i=1 x−di = n∑ i=1 xdi+1 + n∑ i=1 xn−di . Therefore, the terms of the degree sequence of GG are exactly the elements of the set {di + 1 : 1 ≤ i ≤ n} ∪ {n− di : 1 ≤ i ≤ n}. Theorem 11. Let G and H be non-trivial connected graphs. Then fG⊠H(x) = ∑ v∈V (G) fH(x|NG(v)|+1)x|NG(v)| = ∑ p∈V (H) fG(x |NH(p)|+1)x|NH(p)|. Proof. Let (v, p) ∈ V (G ⊠ H). Let D1 = NG(v) × {p}, D2 = {v} × NH(p), and D3 = NG(v) × NH(p). By definition of strong product of two graphs, it follows that NG⊠H((v, p)) = D1 ∪D2 ∪D3. Hence, |NG⊠H((v, p))| = |NG(v)|+ |NH(p)|+ |NG(v)||NH(p)|. Thus, fG⊠H(x) = ∑ (v,p)∈V (G⊠H) x|NG⊠H(v,p)| = ∑ (v,p)∈V (G⊠H) x|NG(v)|+|NH(p)|+|NG(v)||NH(p)| = ∑ v∈V (G) x|NG(v)| ∑ p∈V (H) x(|NG(v)|+1)|NG(p)| Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1457 = ∑ v∈V (G) x|NG(v)|fH(x|NG(v)|+1). Since ∑ (v,p)∈V (G⊠H) x|NG(v)|+|NH(p)|+|NG(v)||NH(p)| = ∑ p∈V (H) x|NH(p)| ∑ v∈V (G) x(|NH(p)|+1)|NG(v)| = ∑ p∈V (H) x|NH(p)|fG(x |NH(p)|+1), it follows that fG⊠H(x) = ∑ v∈V (G) fH(x|NG(v)|+1)x|NG(v)| = ∑ p∈V (H) fG(x |NH(p)|+1)x|NH(p)|. Corollary 6. Let G and H be non-trivial r1-regular and r2-regular connected graphs of orders m and n, respectively. Then fG⊠H(x) = mnxr1+r2+r1r2 . Proof. Since G andH are, respectively, r1-regular and r2-regular graphs, fG(x) = mxr1 and fH(x) = nxr2 . It follows that fH(xr1+1) = nxr2(r1+1). Thus, by Theorem 6, fG⊠H(x) = ∑ v∈V (G) x|NG(v)|fH(x|NG(v)|+1) = ∑ v∈V (G) xr1nxr1r2+r2 = n ∑ v∈V (G) xr1+r2+r1r2 = mnxr1+r2+r1r2 . Theorem 12. Let G and H be non-trivial connected graphs of orders m and n, respec- tively. Then fG⊕H(x) = ∑ v∈V (G) xn|NG(v)|fH(xm−2|NG(v)|). Proof. Let (v, p) ∈ V (G⊕H). From the definition of G⊕H, it follows that NG⊕H((v, p)) = [NG(v)× (V (H) \NH(p)] ∪ [(V (G) \NG(v))×NH(p)]. Hence, |NG⊕H((v, p))| = |NG(v)|(n− |NH(p)|) + |NH(p)|(m− |NG(v)|). Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1458 Therefore, fG⊕H(x) = ∑ (v,p)∈V (G⊕H) x|NG⊕H(v,p)| = ∑ (v,p)∈V (G⊕H) x|NG(v)|(n−|NH(p)|)+|NH(p)|(m−|NG(v)|) = ∑ v∈V (G) ∑ p∈V (H) x|NG(v)|(n−|NH(p)|)+|NH(p)|(m−|NG(v)|) = ∑ v∈V (G) xn|NG(v)| ∑ p∈V (H) x(m−2|NG(v)|)|NH(p)| = ∑ v∈V (G) xn|NG(v)|fH(xm−2|NG(v)|). Corollary 7. Let G and H be non-trivial of orders m and n, respectively. If G is r-regular, then fG⊕H(x) = mxnrfH(xm−2r). Proof. Since G is r-regular, |NG(v)| = r for all v ∈ V (G). Thus, from Theorem 12, we have fG⊕H(x) = ∑ v∈V (G) xn|NG(v)|fH(xm−2|NG(v)|) = ∑ v∈V (G) xnrfH(xm−2r) = mxnrfH(xm−2r). Theorem 13. Let G and H be non-trivial connected graphs of orders m and n, respec- tively. If ⟨d1, d2, · · · dm⟩ and ⟨q1, q2, · · · qn⟩ are the degree sequences of G and H, re- spectively, then the terms of the degree sequence of G ⊕ H are the elements of the set {ndi + (m− 2di)qj : 1 ≤ i ≤ m and 1 ≤ j ≤ n}. Proof. From Theorem 12 and the polynomial representation fH(x) = n∑ j=2 xqj , we have fG⊕H(x) = ∑ v∈V (G) xn|NG(v)|fH(xm−2|NG(v)|) = m∑ i=1 xndi n∑ j=1 x(m−2di)qj Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1459 = m∑ i=1 n∑ j=1 xndix(m−2di)qj = m∑ i=1 n∑ j=1 xndi+(m−2di)qj . It follows that the terms of the degree sequence of G ⊕ H are the elements of the set {ndi + (m− 2di)qj : 1 ≤ i ≤ m and 1 ≤ j ≤ n}. Theorem 14. Let G and H be non-trivial connected graphs of orders m and n, respec- tively. Then fG∨H(x) = fG(x n)fH(xm). Proof. Let (v, p) ∈ V (G ∨H). From the definition of G ∨H, it follows that NG∨H((v, p)) = (NG(v)× V (H)) ∪ (V (G)×NH(p)). Hence, |NG∨H((v, p))| = n|NG(v)|+m|NH(p)|. Therefore, fG∨H(x) = ∑ (v,p)∈V (G∨H) x|NG∨H(v,p)| = ∑ (v,p)∈V (G∨H) xn|NG(v)|+m|NH(p)| = ∑ v∈V (G) xn|NG(v)| ∑ p∈V (H) xm|NH(p)| = ∑ v∈V (G) xn|NG(v)|fH(xm) = fG(x n)fH(xm). Corollary 8. Let n and m be positive integers. Then (i) fPm∨Pn(x) = (m− 2)(n− 2)x2m+2n + 2(m− 2)xm+2n + 2(n− 2)x2m+n + 4xm+n for m,n ≥ 2; (ii) fPm∨Cn(x) = n(m− 2)x2m+2n + 2nx2m+n for m ≥ 2 and n ≥ 3; and (iii) fCm∨Cn(x) = mnx2m+2n for m,n ≥ 3. Proof. For any k ≥ 2 and r ≥ 3, fPk (x) = (k − 2)x2 + 2x and fCr(x) = rx2. By Theorem 14, we have fPm∨Pn(x) = fPm(x n)fPn(x m) Jayhan Cruz, G. Malacas, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1449-1462 1460 = [(m− 2)(xn)2 + 2(xn)][(n− 2)(xm)2 + 2(xm)] = (m− 2)(n− 2)x2m+2n + 2(m− 2)xm+2n + 2(n− 2)x2m+n + 4xm+n, fPm∨Cn(x) = fPm(x n)fCn(x m) = [(m− 2)(xn)2 + 2(xn)](n(xm)2) = n(m− 2)x2m+2n + 2nx2m+n, and fCm∨Cn(x) = fCm(x n)fCn(x m) = (m(xn)2)(n(xm)2) = mnx2m+2n. Theorem 15. Let G and H be non-trivial connected graphs of orders m and n, respec- tively. Then a ∈ R is a zero of fG∨H(x) if and only if an is a zero of fG(x) or am is a zero of fH(x). Proof. By Theorem 14, fG∨H)(x) = fG(x n)fH(xm). Suppose a is a zero of fG∨H)(x). Then fG∨H)(a) = fG(a n)fH(am) = 0. This implies that fG(a n) = 0 or fH(am) = 0. Hence, an is a zero of fG(x) or a m is a zero of fH(x). Conversely, suppose an is a zero of fG(x) or am is a zero of fH(x). Then, clearly, fG∨H)(a) = fG(a n)fH(am) = 0, showing that a is a zero of fG∨H)(x). Theorem 16. Let G and H be non-trivial connected graphs of orders m and n, respec- tively. If ⟨d1, d2, · · · dm⟩ and ⟨q1, q2, · · · qn⟩ are the degree sequences of G and H, re- spectively, then the terms of the degree sequence of G ∨ H are the elements of the set {ndi +mqj : 1 ≤ i ≤ m and 1 ≤ j ≤ n}. Proof. From Theorem 14, fG∨H(x) = fG(x n)fH(xm) = m∑ i=1 xndi n∑ j=1 xmqj = m∑ i=1 n∑ j=1 xndixmqj = m∑ i=1 n∑ j=1 xndi+mqj . It follows that the terms of the degree sequence of G ∨ H are the elements of the set {ndi +mqj : 1 ≤ i ≤ m and 1 ≤ j ≤ n}. REFERENCES 1461 4. Conclusion and Recommendation The polynomial representations of some graphs have been obtained in this study. The authors were not able to describe the degree sequence of some graphs resulting from some operations. However, for particular graphs, the degree sequence of the graphs may be obtained. Determining real roots or zeros of the polynomial representation of a graph, if they exist, can be an aspect for further investigation. 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