EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6048 ISSN 1307-5543 – ejpam.com Published by New York Business Global Independent Neighborhood Polynomial of the Direct and Corona Products of Trees Normalah S. Abdulcarim1,∗, Susan C. Dagondon2 1 Department of Mathematics, College of Natural Sciences and Mathematics, Mindanao State University Main Campus, 9700 Marawi City, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra and Analysis-Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Philippines Abstract. Let G be a connected graph. We say that a given graph is a tree if every pair of vertices is connected by a unique path. The corona product of two graphs G and H is defined as the graph obtained by taking one copy of G and |V (G)| copies of H and joining the ith vertex of G to every vertex in the ith copy of H. On the other hand, the direct product G × H is a graph such that the vertex set of G×H is the Cartesian product V (G)×V (H); and vertices (g, h) and (g′, h′) are adjacent in G×H if and only if g is adjacent to g′ in G, and h is adjacent to h′ in H. Here, authors worked on the independent neighborhood sets of the direct and corona products of two trees are obtained. Also, corresponding independent neighborhood polynomialsare found. 2020 Mathematics Subject Classifications: 05C05, 05C31, 05C76 Key Words and Phrases: Corona Product, Direct Product, Independent Neighborhood Poly- nomial 1. Introduction A graph G is a pair (V (G), E(G)) consisting of a nonempty finite set of vertices V (G) and a set of edges E(G) of unordered pairs of elements of V (G). The cardinalities of V (G) and E(G) are called the order and size of G, respectively. We write x = uv and say that u and v are adjacent vertices; vertex u and edge x are incident with each other, so are v and x. The two vertices incident with an edge are its end vertices or ends, and an edge joins its ends. Two vertices of a graph G are said to be neighbors if they are adjacent in G. The neighborhood of a vertex v ∈ V (G) is the set N(v) = {w : w ∈ V (G) and vw ∈ E(G)}. A vertex v is pendant if its neighborhood contains only one vertex; and edge e = uv is pendant if one of its end vertices is a pendant vertex. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6048 Email addresses: normalah.abdulcarim@msumain.edu.ph (N. Abdulcarim), susan.dagondon@g.msuiit.edu.ph (S. Dagondon) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 2 of 14 A graph is acyclic if it has no cycles. A graph is said to be connected if every pair of vertices are joined by a path. A graph that is not connected is called disconnected. A tree is a connected acyclic graph. In 2008, Brown et al. [1] studied the neighborhood polynomial of a graph. Later, in 2014, Alwardi et al. [2] also investigated the neighborhood polynomial of graphs. The following year, in 2015, Kulli [3] focused on the neighborhood graph of a graph. Motivated by these studies, the authors draw inspiration from Murthy’s paper ”The Independent Neighborhood Polynomial of Graphs.” [4] The main objective of the present paper is to establish results on the independent neighborhood polynomial for the direct and corona products of two trees. 2. Preliminaries This section presents some basic concepts and known results needed in this study. Definition 1. [5] A graph is acyclic if it has no cycles. A graph is said to be connected if every pair of vertices are joined by a path. A graph that is not connected is called disconnected. A tree is a connected acyclic graph. Definition 2. [6] Let G be a graph. The distance between two vertices x and y in a graph G, denoted by dG(x, y) or simply d(x, y), is the length of the shortest path joining them, otherwise, d(x, y) = ∞. Theorem 1. [7] Let G be any tree. Then for any u ∈ V (G), the set Ω = {v ∈ V (G) : d(u, v) = 2n, n ∈ N∗} is an independent neighborhood set of G. Corollary 1. [7] Let F be the set of nonpendant vertices in a tree G and f ∈ F . Let A = {a ∈ F : dG(a, f) = 2n, n ∈ N∗} and B = F \A. Then the sets Ω = {v : v is a pendant neighbor of a, for some a ∈ A} ∪B and ∆ = {u : u is a pendant neighbor of b, for some b ∈ B} ∪A are the only independent neighborhood sets of G. Remark 1. [7] For any tree T with independent neighborhood sets Ω and ∆, Ni(T, x) = x|Ω| + x|∆|. N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 3 of 14 Corollary 2. [7] For any tree T with independent neighborhood sets Ω and ∆, if uv ∈ E(T ), then u ∈ Ω and v ∈ ∆. Definition 3. [8] The direct product (or conjunction) of two graphs G and H, denoted by G · H, is the graph whose vertex set is V (G) × V (H) in which (u, v) is adjacent to (u′, v′) if and only if uu′ ∈ E(G) and vv′ ∈ E(H). Example 1. Figure 1 shows two graphs G and H and their direct product. a b c d e G : 1 2 4 3 H : (a, 1) (a, 4) (a, 3) (b, 2) (c, 2) (d, 2) (e, 1) (e, 3) (e, 4) G ·H : (b, 1) (b, 4) (b, 3) (c, 1) (c, 3) (c, 4) (d, 1) (d, 3) (d, 4) (a, 2) (e, 2) Figure 1: Graphs G and H and their direct product Definition 4. [5] The corona of two graphs G and H, denoted G ◦H, is defined as the graph obtained by taking one copy of G and |V (G)| copies of H and joining the ith vertex of G to every vertex in the ith copy of H. It is customary to denote by Hv that copy of H whose vertices are adjoined with the vertex v of G. In effect, G ◦H is composed of the subgraphs Hv + v joined together by the edges of G. Moreover, V (G ◦H) = ⋃ v∈V (G) V (Hv + v). Example 2. Figure 2 shows two graphs G and H and their corona products G ◦H and H ◦G. N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 4 of 14 1 2 3 4 5 G a b c H 4a 4b 4c 4 5a 5b 5c 5 3a 3b 3c 3 2a 2b 2c 2 1a 1b 1c 1 G ◦H a a1 a2 a3 a4 a5 b b1 b2 b3 b4 b5 c c1 c2 c3 c4 c5 H ◦G Figure 2: Two trees and their two coronas Definition 5. [9] The open neighborhood of a vertex x, denoted byN(x), is a set containing all vertices y which are adjacent to x, that is, N(x) = {y ∈ V (G) : xy ∈ E(G)}. In case N(x) is a singleton, x is an end-vertex. The closed neighborhood of a vertex x of G is the set N [x] = N(x) ∪ {x}. Definition 6. [4] A set S of vertices in a graph G is a neighborhood set if G = ⋃ v∈S ⟨N [v]⟩ where ⟨N [v]⟩ is the subgraph of G induced by v and all the vertices adjacent to v. The neighborhood number of G is the minimum cardinality of neighborhood sets, denoted by ni(G). Example 3. Consider the graph H in Figure 3. v1 v2 v3 v4v5 H : Figure 3: A graph with neighborhood number equals 2 The neighborhood sets ofH are {v2, v4}, {v1, v3, v5}, {v1, v2, v4}, {v1, v2, v3, v4}, {v1, v2, v3, v5}, N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 5 of 14 {v1, v2, v4, v5}, {v1, v3, v4, v5}, {v2, v3, v4, v5} and {v1, v2, v3, v4, v5}. Here, ni(H) = 2. Definition 7. [4] A set S ⊆ V (G) is an independent neighborhood set of G, if S is a neighborhood set and no two vertices in S are adjacent. Remark 2. Not all graphs have independent neighborhood set. If it has an independent neighborhood set, then it is called an independent neighborhood graph or an IN -graph. 1 5 4 3 2 Figure 4: A graph that is not an IN -graph Definition 8. [4] Let G = (V,E) be a graph with m vertices. Then the independent neighborhood polynomial of G of order m is Ni(G, x) = m∑ j=ηi(G) ni(G, j)xj , where ni(G, j) is the number of independent neighborhood sets of G of cardinality j and ηi(G) is the minimum cardinality of an independent neighborhood set which is called the independent neighborhood number of G. Example 4. In Figure 3, the only independent neighborhood sets of H are {v2, v4} and {v1, v3, v5}. Therefore, the independent neighborhood polynomial of H is Ni(H,x) = x2 + x3. Proposition 1. [4] Let G = G1∪G2 where G1, G2 be any two IN -graphs. Then Ni(G, x) = Ni(G1, x)Ni(G2, x). Proposition 2. [4] Let G be an IN -graph with n+1 vertices. Then Ni(G, x) = x(1+xn−1) if and only if G ∼= k1,n. 3. Main Results 3.1. Independent Neighborhood Polynomial of the Direct Product of Trees Theorem 2. For any trees G and H, the direct product of G and H, G·H, is disconnected N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 6 of 14 Proof. Label the vertices of G by xi and of H by yj . Since G and H are trees, by Corollary 1, G and H have independent neighborhood sets say Ω1,∆1 and Ω2,∆2, respec- tively. We now show that G ·H is disconnected, that is, there exist (xi1 , yj1), (xi2 , yj2) ∈ V (G · H) such that (xi1 , yj1) and (xi2 , yj2) are not connected by any path. Consider (x1, y1), (x2, y2) ∈ V (G · H) such that x1x2 /∈ E(G) or y1y2 /∈ E(H). Without loss of generality, let xi1 , xi2 ∈ Ω1, yj1 ∈ Ω2 and yj2 ∈ ∆2. Clearly, (xi1 , yj1)(xi2 , yj2) /∈ E(G ·H) since xi1 , xi2 ∈ Ω2 which means xi1 and xi2 are nonadjacents. Observe that for any (xi, yi)- (xn, yn) path in G ·H, if xi ∈ Ω1, yi ∈ Ω2, then xi+1 ∈ ∆1, yi+1 ∈ ∆2 while if xi ∈ Ω1 and yi ∈ ∆2, then xi+1 ∈ ∆1 and yi+1 ∈ Ω2. Now, since xi1 ∈ Ω1, yj1 ∈ Ω2 and yj2 ∈ ∆2 but xi2 ∈ ∆1, it follows that there is no path connecting the vertices (xi1 , yj1) and (xi2 , yj2) in G ·H. Therefore, G ·H is disconnected. Theorem 3. For any trees G and H, the direct product of G and H has 2 components, say A and B, that is, G ·H = A ∪B. Proof. Let G and H be any trees with independent neighborhood sets Ω1,∆1 and Ω2,∆2, respectively. Assume that G · H has more than 2 components, say Ar, r ∈ Z+. This means that for every (ur, vr) ∈ Ar, (ur, vr) are disconnected ∀r. We note that for any (u, v) ∈ V (G ·H), either u ∈ Ω1 or u ∈ ∆1 and v ∈ Ω2 or v ∈ ∆2. Furthermore, for any (ui, vi)-(un, vn) path in G · H, uiui+1 ∈ E(G) and vivi+1 ∈ E(H). By Corollary 2, if uiui+1 ∈ E(G) and vivi+1 ∈ E(H), then u1 ∈ Ω1, ui+1 ∈ ∆1 and v1 ∈ Ω2, vi+1 ∈ ∆2. It follows that for (ui, vi)-(un, vn) path in G · H, if ui ∈ Ω1, vi ∈ Ω2 for i odd, then uj ∈ ∆1, vj ∈ ∆2 for j even while if ui ∈ Ω1, vi ∈ ∆2, then uj ∈ ∆1, vj ∈ Ω2. Now, consider (u1, v1) ∈A1 such that u1 ∈ Ω1, v1 ∈ Ω2, (u2, v2) ∈A2 such that u2 ∈ Ω1, v2 ∈ ∆2, (u3, v3) ∈A3 such that u3 ∈ ∆1, v3 ∈ Ω2, (u4, v4) ∈A4 such that u4 ∈ ∆1, v4 ∈ ∆2. We can easily see that (u1, v1) and (u3, v3) are disconnected, so are (u1, v1) and (u2, v2), (u2, v2) and (u4, v4), and (u3, v3) and (u4, v4). But observe that (u1, v1)-(u4, v4) is a path in G ·H. This implies (u1, v1) and (u4, v4) belong to 1 component. Similarly, (u2, v2) and (u3, v3) belong to 1 component. This is a contradiction. Thus, G · H cannot have more than 2 components. Therefore, G ·H has exactly 2 components. Theorem 4. Suppose G has independent neighborhood sets Ω1,∆1 and H has independent neighborhood sets Ω2,∆2. If A and B are the components of G ·H, then V (A) = {(x, y) : x ∈ Ω1, y ∈ Ω2} ∪ {(w, z) : w ∈ ∆1, z ∈ ∆2} and V (B) = {(x, y) : x ∈ Ω1, y ∈ ∆2} ∪ {(w, z) : w ∈ ∆1, z ∈ Ω2}. N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 7 of 14 Proof. The proof is similar to the proof of Theorem 3. Corollary 3. Let A and B be the components of G ·H such that V (A) = {(x, y) : x ∈ Ω1, y ∈ Ω2} ∪ {(w, z) : w ∈ ∆1, z ∈ ∆2} and V (B) = {(x, y) : x ∈ Ω1, y ∈ ∆2} ∪ {(w, z) : w ∈ ∆1, z ∈ Ω2}. Then the independent neighborhood sets of A are the sets {(x, y) : x ∈ Ω1, y ∈ Ω2} and {(w, z) : w ∈ ∆1, z ∈ ∆2} while the independent neighborhood sets of B are the sets {(x, y) : x ∈ Ω1, y ∈ ∆2} and {(w, z) : w ∈ ∆1, z ∈ Ω2} where Ω1,∆1 are the independent neighborhood sets of G and Ω2,∆2 are the independent neighborhood sets of H. Proof. Let A and B be the components of G ·H with V (A) = {(x, y) : x ∈ Ω1, y ∈ Ω2} ∪ {(w, z) : w ∈ ∆1, z ∈ ∆2} and V (B) = {(x, y) : x ∈ Ω1, y ∈ ∆2} ∪ {(w, z) : w ∈ ∆1, z ∈ Ω2}. Suppose A1 = {(x, y) : x ∈ Ω1, y ∈ Ω2}, A2 = {(w, z) : w ∈ ∆1, z ∈ ∆2}, B1 = {(x, y) : x ∈ Ω1, y ∈ ∆2} and B2 = {(w, z) : w ∈ ∆1, z ∈ Ω2}. We will show that A1, A2 are the independent neighborhood sets of A and B1, B2 are the in- dependent neighborhood sets of B. Consider A1. Let (x1, y1), (x2, y2) ∈ V (A1). Since x1, x2 ∈ Ω1, y1, y2 ∈ Ω2 where Ω1 and Ω2 are the independent neighborhood sets of G and H, respectively, implies x1, x2 are nonadjacent vertices, so are y1, y2. Thus, (x1, y1)(x2, y2) /∈ E(A). This means (x1, y1) and (x2, y2) are nonadjacent vertices. Next, we will show that ⋃ (x,y)∈A1 ⟨N [(x, y)]⟩ = A. Assume to the contrary that⋃ (x,y)∈A1 ⟨N [(x, y)]⟩ ≠ A. Then there exists (x3, y3)(x4, y4) ∈ E(A) such that (x3, y3)(x4, y4) /∈⋃ (x,y)∈A1 ⟨N [(x, y)]⟩. It follows that both (x3, y3), (x4, y4) /∈ V (A1). Hence, x3, x4 ∈ ∆1 and y3, y4 ∈ ∆2. Since both ∆1 and ∆2 are independent neighborhood sets, x3 and x4 are nonadjacents, so are y3 and y4. Thus, x3x4 /∈ E(G) and y3y4 /∈ E(H). It follows that (x3, y3)(x4, y4) /∈ E(G·H) which is a contradiction. Hence, ⋃ (x,y)∈A1 ⟨N [(x, y)]⟩ = A. There- fore, A1 is an independent neighborhood set of A. Similarly, we can show that A2 is also an independent neighborhood set of A. Following the same argument for B1 and B2, we have B1 and B2 are the independent neighborhood sets of B. Corollary 4. Let G and H be trees with independent neighborhood sets Ω1,∆1 and Ω2,∆2, respectively. Then Ni(G ·H,x) = x|Ω1||Ω2|+|Ω1||∆2| + x|Ω1||Ω2|+|∆1||Ω2| + x|∆1||∆2|+|Ω1||∆2| + x|∆1||∆2|+|∆1||Ω2|. N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 8 of 14 Proof. Suppose G and H are trees with independent neighborhood sets Ω1,∆1 and Ω2,∆2, respectively. By Theorems 3 and 4, G ·H = A ∪B where V (A) = {(x, y) : x ∈ Ω1, y ∈ Ω2} ∪ {(w, z) : w ∈ ∆1, z ∈ ∆2} and V (B) = {(x, y) : x ∈ Ω1, y ∈ ∆2} ∪ {(w, z) : w ∈ ∆1, z ∈ Ω2}. By Corollary 3, the sets {(x, y) : x ∈ Ω1, y ∈ Ω2} and {(w, z) : w ∈ ∆1, z ∈ ∆2} are the independent neighborhood sets of A while the sets {(x, y) : x ∈ Ω1, y ∈ ∆2} and {(w, z) : w ∈ ∆1, z ∈ Ω2} are the independent neighborhood sets of B. We note that |{(x, y) : x ∈ Ω1, y ∈ Ω2}| =|Ω1||Ω2| |{(w, z) : w ∈ ∆1, z ∈ ∆2}| =|∆1||∆2| |{(x, y) : x ∈ Ω1, y ∈ ∆2}| =|Ω1||∆2| |{(w, z) : w ∈ ∆1, z ∈ Ω2}| =|∆1||Ω2|. Thus, Ni(A, x) = x|Ω1||Ω2| + x|∆1||∆2| and Ni(B, x) = x|Ω1||∆2| + x|∆1||Ω2|. Since G ·H = A ∪B, by Proposition 1, Ni(G ·H,x) = Ni(A, x)Ni(B, x). Therefore, Ni(G ·H,x) = ( x|Ω1||Ω2| + x|∆1||∆2| )( x|Ω1||∆2| + x|∆1||Ω2| ) =x|Ω1||Ω2|x|Ω1||∆2| + x|Ω1||Ω2|x|∆1||Ω2| + x|∆1||∆2|x|Ω1||∆2| + x|∆1||∆2|x|∆1||Ω2| =x|Ω1||Ω2|+|Ω1||∆2| + x|Ω1||Ω2|+|∆1||Ω2| + x|∆1||∆2|+|Ω1||∆2| + x|∆1||∆2|+|∆1||Ω2|. Example 5. Given the graphs G and H and their corresponding direct product in Figure 5. We note that the independent neighborhood sets of G are Ω1 = {a, c, d} and ∆1 = {b, e, f} while the independent neighborhood sets of H are Ω2 = {1, 3, 4} and ∆2 = N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 9 of 14 a b c d e f G : 1 2 3 4 5 6 7 H : (a, 1) (a, 3) (a, 4) (b, 2) (b, 5) (b, 6) (b, 7) (c, 1) (c, 3) (c, 4) (d, 1) (d, 3) (d, 4) (e, 2) (e, 5) (e, 6) (e, 7) (f, 2) (f, 5) (f, 6) (f, 7) A : (a, 2) (a, 5) (a, 6) (a, 7) (b, 1) (b, 3) (b, 4) (c, 2) (c, 5) (c, 6) (c, 7) (d, 2) (d, 5) (d, 6) (d, 7) (e, 1) (e, 3) (e, 4) (f, 1) (f, 3) (f, 4) B : G ·H Figure 5: The graphs G and H and their direct product G ·H {2, 5, 6, 7}. By Corollary 3, the sets {(x, y) : x ∈ Ω1, y ∈ Ω2} = {(a, 1), (a, 3), (a, 4), (c, 1), (c, 3), (c, 4), (d, 1), (d, 3), (d, 4)} and {(w, z) : w ∈ ∆1, z ∈ ∆2} ={(b, 2), (b, 5), (b, 6), (b, 7), (e, 2), (e, 5), (e, 6), (e, 7), (f, 2), (f, 5), (f, 6), (f, 7)} are the independent neighborhood sets of A while the sets {(x, y) : x ∈ Ω1, y ∈ ∆2} ={(a, 2), (a, 5), (a, 6), (a, 7), (c, 2), (c, 5), (c, 6), (c, 7), (d, 2), (d, 5), (d, 6), (d, 7)} and {(w, z) : w ∈ ∆1, z ∈ Ω2} = {(b, 1), (b, 3), (b, 4), (e, 1), (e, 3), (e, 4), (f, 1), (f, 3), (f, 4)} N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 10 of 14 are the independent neighborhood sets of B. Furthermore, |Ω1| = 3, |∆1| = 3, |Ω2| = 3 and |∆2| = 4. Therefore, by Corollary 4, Ni(G ·H,x) =x3(3)+3(4) + x3(3)+3(3) + x3(4)+3(4) + x3(4)+3(3) =x21 + x18 + x24 + x21 =x18 + 2x21 + x24. Theorem 5. For any graphs G and H, the direct product of G and H is commutative, that is, G ·H = H ·G. Proof. Let G and H be any graphs. To show that the direct product of G and H is commutative, we will show that G ·H and H ·G are isomorphic. We note that |V (G ·H)| = |V (G)×V (H)| = |V (H)×V (G)| = |V (H ·G)|. Define the map β : V (G ·H) −→ V (H ·G) by β((g, h)) = (h, g) for all (g, h) ∈ V (G ·H). For (g1, h1), (g2, h2) ∈ V (G ·H) such that β((g1, h1)) = β((g2, h2)), we have (h1, g1) = β((g1, h1)) = β((g2, h2)) = (h2, g2) =⇒ (h1, g1) = (h2, g2) =⇒ h1 = h2 and g1 = g2. Thus, (g1, h1) = (g2, h2) which shows β is one-to-one. Now, observe that for every (h, g) ∈ V (H ·G), there exists (g, h) ∈ V (G·H) such that β((g, h)) = (h, g), for all (g, h) ∈ V (G·H). This implies β(V (G · H)) = V (H · G) and it follows that β is onto. Finally, we let (g1, h1), (g2, h2) ∈ V (G · H). Then (g1, h1)(g2, h2) ∈ E(G · H) ⇔ g1g2 ∈ E(G) and h1h2 ∈ E(H) ⇔ (h1, g1)(h2, g2) ∈ E(H ·G). Hence, β preserves adjacency. Consequently, G ·H ∼= H ·G. Therefore, G ·H = H ·G. Remark 3. For any trees G and H, the independent neighborhood poly- nomial of G ·H is the same as the independent neighborhood polynomial of H ·G, that is Ni(G ·H,x) = Ni(H ·G, x). 3.2. Independent Neighborhood Polynomial of the Corona Product of Two Trees Theorem 6. Let G and H be any trees. Suppose G has independent neighborhood sets Ω1,∆1 and H has independent neighborhood sets Ω2,∆2. Then Ni(G ◦H,x) = x|∆1| ( x|Ω2| + x|∆2| )|Ω1| + x|Ω1| ( x|Ω2| + x|∆2| )|∆1| . N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 11 of 14 Proof. Let G and H be any trees. Suppose G and H has independent neighborhood sets Ω1,∆1 and Ω2,∆2, respectively. Then by Remark 1, Ni(G, x) = x|Ω1| + x|∆1| and Ni(H,x) = x|Ω2| + x|∆2|. By definition of corona of two graphs, every ith vertex of G is joined to every vertex in the ith copy of H. Observe that the independent neighborhood set Ω1 of G and the independent neighborhood sets in each copy of H joined to every vertex in ∆1 forms an independent neighborhood set in G ◦ H. Similarly, the independent neighborhood set ∆1 of G and the independent neighborhood sets in each copy of H joined to every vertex in Ω1 is also an independent neighborhood set in G ◦H. Moreover, the number of combinations of the indepedent neighborhood sets of each copy ofH joined to every u ∈ ∆1 is |∆1|, that is, Ni  ⋃ j∈∆1 Hj , x  = ( x|Ω2| + x|∆2| )|∆1| . Also, there are |Ω1| combinations of independent neighborhood sets of each copy of H joined to each v ∈ Ω1, that is, Ni  ⋃ r∈Ω1 Hr, x  = ( x|Ω2| + x|∆2| )|Ω1| . Hence, Ni(G ◦H,x) = x|∆1| ( x|Ω2| + x|∆2| )|Ω1| + x|Ω1| ( x|Ω2| + x|∆2| )|∆1| . Example 6. Consider the corona of two graphs B(2, 2) and K1,3. 1 2 3 4 5 6 B(2, 2) a b c d K1,3 a1 b1 c1 d1 1 a2 b2 c2 d2 2 a3 b3 c3 d3 3 a4 b4 c4 d4 4 a5 b5 c5 d5 5 a6 b6 c6 d6 6 B(2, 2) ◦K1,3 Figure 6: The graph B(2, 2),K1,3 and B(2, 2) ◦K1,3 N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 12 of 14 Note that G has independent neighborhood sets Ω1 = {1, 2, 4},∆1 = {3, 5, 6} and H has independent neighborhood sets Ω2 = {d}, ∆2 = {a, b, c}. We can see that the inde- pendent neighborhood sets of G ◦H are {1, 2, 4, a3, b3, c3, a5, b5, c5, a6, b6, c6}, {1, 2, 4, a3, b3, c3, a5, b5, c5, d6}, {1, 2, 4, a3, b3, c3, d5, a6, b6, c6}, {1, 2, 4, a3, b3, c3, d5, d6}, {1, 2, 4, d3, a5, b5, c5, a6, b6, c6}, {1, 2, 4, d3, a5, b5, c5, d6}, {1, 2, 4, d3, d5, a6, b6, c6}, {1, 2, 4, d3, d5, d6}, {3, 5, 6, a1, b1, c1, a2, b2, , c2, a4, b4, c4}, {3, 5, 6, a1, b1, c1, a2, b2, , c2, d4}, {3, 5, 6, a1, b1, c1, d2, a4, b4, c4}, {3, 5, 6, a1, b1, c1, d2, d4}, {3, 5, 6, d1, a2, b2, , c2, a4, b4, c4}, {3, 5, 6, d1, d2, a4, b4, c4}, {3, 5, 6, d1, a2, b2, , c2, d4}, {3, 5, 6, d1, d2, d4}. Moreover, by Theo- rem 6, Ni(G ◦H,x) = x|∆1| ( x|Ω2| + x|∆2| )|Ω1| + x|Ω1| ( x|Ω2| + x|∆2| )|∆1| = x3(x+ x3)3 + x3(x+ x3)3 = 2x3(x+ x3)3 = 2x3(x3 + 3x5 + 3x7 + x9) = 2x6 + 6x8 + 6x10 + 2x12. On the other hand, the corona H ◦G is given in Figure 7. 1a 2a 3a 4a 5a 6a a 1b 2b 3b 4b 5b 6b b 1c 2c 3c 4c 5c 6c c 1d 2d 3d 4d 5d 6d d K1,3 ◦B(2, 2) Figure 7: The graph K1,3 ◦B(2, 2) We can see that the independent neighborhhood sets ofK1,3◦B(2, 2) are {a, b, c, 1d, 2d, 4d}, {a, b, c, 3d, 5d, 6d}, {d, 1a, 2a, 4a, 1b, 2b, 4b, 1c, 2c, 4c}, {d, 1a, 2a, 4a, 1b, 2b, 4b, 3c, 5c, 6c}, {d, 1a, 2a, 4a, 3b, 5b, 6b, 1c, 2c, 4c}, {d, 1a, 2a, 4a, 3b, 5b, 6b, 3c, 5c, 6c}, {d, 3a, 5a, 6a, 1b, 2b, 4b, 1c, 2c, 4c}, {d, 3a, 5a, 6a, 1b, 2b, 4b, 3c, 5c, 6c}, {d, 3a, 5a, 6a, 3b, 5b, 6b, 1c, 2c, 4c}, {d, 3a, 5a, 6a, 3b, 5b, 6b, 3c, 5c, 6c}. By Theorem 6, Ni(K1,3 ◦B(2, 2)) = x3(x3 + x3) + x(x3 + x3)3 N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 13 of 14 = x3(2x3) + x(2x3)3 = 2x6 + 8x10. Theorem 7. For any trees G and H, if G ∼= H, then Ni(G ◦H,x) = Ni(H ◦G, x). Proof. Let G and H be any trees. Suppose G ∼= H. Then G and H has indepedent neighborhood sets Ω1,∆1 and Ω2,∆2, respectively. Since G ∼= H, either |Ω1| = |Ω2| and |∆1| = |∆2| or |Ω1| = |∆2| and |∆1| = |Ω2|. Without loss of generality, assume |Ω1| = |Ω2| and |∆1| = |∆2|. Thus, Ni(G ◦H,x) = x|∆1| ( x|Ω2| + x|∆2| )|Ω1| + x|Ω1| ( x|Ω2| + x|∆2| )|∆1| = x|∆2| ( x|Ω1| + x|∆1| )|Ω2| + x|Ω2| ( x|Ω1| + x|∆1| )|∆2| = Ni(H ◦G, x). Example 7. Consider the corona of K1,4 to itself as shown in Figure 8. 1 2 3 4 0 K1,4 : b c d e a K1,4 : b1 c1 d1 e1 a1 1 b2 c2 d2 e2 a2 2 b3 c3 d3 e3 a3 3 b4 c4 d4 e4 a4 4 b0 c0 d0 e0 a0 0 K1,4 ◦K1,4 Figure 8: The graph K1,4 ◦K1,4 By Proposition 2, K1,4 has independent neighborhood sets of cardinalities 1 and 4, N. Abdulcarim, S. Dagondon / Eur. J. Pure Appl. Math, 18 (3) (2025), 6048 14 of 14 respectively. This implies |Ω1| = 1, |∆1| = 4, |Ω2| = 1, and |∆2| = 4. Therefore, Ni(K1,4 ◦K1,4, x) = x1(x+ x4)4 + x4(x+ x4)1 = x(x4 + 4x3 · x4 + 6x2 · x8 + 4x · x12 + x16) + x5 + x8 = 2x5 + 5x8 + 6x11 + 4x14 + x17. Acknowledgements This research is funded by the Department of Science and Technology (DOST), Min- danao State University Iligan Institute of Technology, and the Mindanao State University Main Campus, Marawi City. References [1] J.I. Brown and R.J. Nowakowski. The neighbourhood polynomial of a graph. As- tralasian Journal of Combinatorics, 42:55–68, 2008. [2] A. Alwardi and P.M. Shivaswamy. On the neighbourhood polynomial of graphs. Eu- ropean Journal of Pure and Applied Mathematics, 6:13–24, 2016. [3] V.R. Kulli. The neighborhood graph of a graph. International Journal of Fuzzy Mathematical Archive, 8:93–99, 2015. [4] Puttaswamy K.B. Murthy. On the independent neighbourhood polynomial of graphs. Indian Streams Research Journal,, pages Vol.5, 1–7, 2014. [5] F. Harary. Graph Theory. Addison-Wesley Publishing Company, 1969. [6] R. Diestel. Graph Theory, Fifth Edition. Springer Nature, 2017. [7] N. Abdulcarim and S. Dagondon. On the independent neighborhood polynomial of the rooted product of two trees. European Journal of Pure and Applied Mathematics, 1:64–81, 2022. [8] J.A. Bondy and U.S.R. Murthy. Graph Theory. Springer, 2007. [9] F. Buckley and F. Harary. Distance in Graphs. Addison-Wesley Series in Mathematics, Reedwood City, 1990.