Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1919 https://internationalpubls.com Even Hamming Distance Labeling of Some Path Related Graphs E.Esakkiammal 1, K.Thirusangu2 and S.Seethalakshmi3 1,2 Department of Mathematics, S.I.V.E.T. College, Gowrivakkam, Chennai, India. 3 Department of Mathematics, R.V. Govt. Arts College, Chengalpattu,Chennai,India. 1esakkiammal2682@gmail.com.2kthirusangu@gmail.com,3seetha0687@gmail.com. Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract Hamming distance is used in error correction while transmitting data over computer networks. In this paper, it is shown that Comb graph 𝑃𝑚 +, Twig graph 𝑇𝑊(𝑃𝑚), Centipede (𝑚, 2) graph, and comb product of 𝑃𝑚 and 𝑃𝑟 graph(𝑃𝑚 ⊳ 𝑃𝑟) are even hamming distance labeled graphs. It is proved that the even hamming distance number of Comb graph and Twig graph are 6 and 8 respectively. Also the even hamming distance number of Centipede graph (m,2) is 6 if m=1 and 8 if 𝑚 > 1 and for the comb product of 𝑃𝑚 and 𝑃𝑟 graph is 4 if 𝑚 = 1 and 6 if 𝑚 > 1 are obtained. This labeling is applied in Cryptography for sharing secret messages. Keywords: Even Hamming distance labeling, Comb graph, Twig graph, Centipede graph and comb product of 𝑃𝑚 and 𝑃𝑟 graph. 1.Introduction Let G be a graph with vertex set V and edge set E. An alternating sequence of vertices and edges, beginning and ending with vertices is called a path graph[3]. A path graph on m+1 vertices is denoted by 𝑃𝑚, 𝑚 ≥ 1 . The Corona of path graph 𝑃𝑚 is obtained from 𝑃𝑚 by attaching a pendent vertex to each vertex of 𝑃𝑚 and it is denoted by 𝑃𝑚 +. This graph is also known as comb graph[2]. A Twig TW(𝑃𝑚), 𝑛 ≥ 2 is a graph obtained from a path by attaching exactly two pendant vertices to each internal vertices of the path[1]. Centipede Graph (m,2) is a graph on 3n vertices obtained by joining two pendant edges that are adjacent in each vertex of a path[6].The comb product between graphs Pm and Pr is a graph obtained by taking one copy of Pm and |V(Pm)| copies of Pr and joining each copy of Pr with each vertex of Pm and this graph is denoted by Pm ⊳ Pr [4]. 2. Even Hamming Distance Labelling of some Graphs 2.1. Definition: Let G = (V, E) be a graph. A function 𝑓:𝑉 → 𝑁 ∪ {0} is said to be an Even hamming distance labeling if there exist an induced function 𝑓∗ ∶ 𝐸 → {2,4,6, … , n} such that for every 𝑢𝑣 ∈ 𝐸, 𝑓∗(𝑢𝑣) = ℎ𝑑([𝑓(𝑢)]2, [𝑓(𝑣)]2) satisfying the following conditions: (i) For every vertex 𝑣 𝜖 𝑉, the set of all edges incident with 𝑣 receive distinct even labels. (ii) For every edge 𝑒 = 𝑢𝑣, the adjacent vertices 𝑢 and 𝑣 receive distinct labels. A graph which admits even hamming distance labeling is called even hamming distance graph. The even hamming distance number of a graph G is the least positive integer n such that 2𝑛 − 1 ≥ 𝑘, where 𝑘 = max {𝑓(𝑣)/𝑣 ∈ 𝑉} and it is denoted by ηℎ𝑑 ′′ (G). mailto:esakkiammal2682@gmail.com mailto:kthirusangu@gmail.com mailto:seetha0687@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1920 https://internationalpubls.com We have proved the existence of even Hamming distance labeling of cycle related graphs in [5]. Here, we have given the proof for existence of even hamming distance labeling of path related graphs. We propose the following algorithms only to label the vertices of the graph. Algorithm 2.1.1: Even hamming Distance Labeling of Comb graph Input: Vertices of 𝑃𝑚 + graph, 𝑚 ≥ 1 𝑉 ← {𝑣𝑖 , 𝑣𝑖 ′/0 ≤ 𝑖 ≤ 𝑚} 𝑣0 ← 0; 𝑣0 ′ ← 15; for 0 ≤ 𝑖 ≤ 𝑚 𝑣𝑖 ← { 3 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 12 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 15 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) ; end for for 0 ≤ 𝑖 ≤ 𝑚-1 𝑣𝑖 , ← { 60 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 51 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 63 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 48 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) ; end for 𝑣𝑚 , ← { 12 𝑖𝑓 𝑚 ≡ 1(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑚 ≡ 2(𝑚𝑜𝑑4) 15 𝑖𝑓 𝑚 ≡ 3(𝑚𝑜𝑑4) 3 𝑖𝑓 𝑚 ≡ 0(𝑚𝑜𝑑4) ; end procedure output: The labeled vertices of 𝑃𝑚 + graph. Theorem: 2.1.2. The corona of path (comb) graph 𝑃𝑚 + is an even hamming distance labeled graph and the even hamming distance number is ηℎ𝑑 ′′ (𝑃𝑚 +) = 6. Proof: Let us consider the comb graph 𝑃𝑚 + with vertex set 𝑉 = {{𝑣0,𝑣1,𝑣2,… , 𝑣𝑚} ∪ {𝑣0, ′ 𝑣1, ′ 𝑣2, ′ … , 𝑣𝑚, ′ }} and edge set 𝐸 = {{𝑣𝑖𝑣𝑖+1 / 0 ≤ 𝑖 ≤ 𝑚 − 1} ∪ {𝑣𝑖𝑣𝑖 ′/0 ≤ 𝑖 ≤ 𝑚}}. Define a function 𝑓: V → N ∪ {0} such that 𝑓(𝑢) ≠ 𝑓(𝑣) for any two adjacent vertices u and v as given in the above algorithm 2.1.1. hence the adjacent vertices receive distinct labels. The edge labels are obtained as follows: 𝑓∗(𝑣0𝑣0 ′) = ℎ𝑑([𝑓(𝑣0)]2, [𝑓(𝑣0 ′)]2) = hd([0]2, [15]2) = hd(00000, 01111) =4. 𝑓∗(𝑣0𝑣1) = ℎ𝑑([𝑓(𝑣0)]2, [𝑓(𝑣1)]2)= = hd([0]2, [3]2 = hd(00000, 00011) = 2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1921 https://internationalpubls.com For 1 ≤ i ≤ m − 1 Case (i): if i ≡ 1(mod 4)or 𝑚 ≡ 1(𝑚𝑜𝑑 4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([3]2, [12]2) = 4. 𝑓∗(𝑣𝑖𝑣𝑖 ′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([3]2, [60]2) = 6. 𝑓∗(𝑣𝑚𝑣𝑚 ′ ) =ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′ )]2)= hd([3]2, [12]2) = 4. Case (ii): if i ≡ 2(mod4); or 𝑚 ≡ 2(𝑚𝑜𝑑 4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([12]2, [0]2) = 2. 𝑓∗(𝑣𝑖𝑣𝑖 ′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd(122, [51]2) = 6. 𝑓∗(𝑣𝑚𝑣𝑚 ′ ) =ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′ )]2)= hd([12]2, [0]2) = 2. Case (iii): if i ≡ 3(mod 4) or 𝑚 ≡ 2(𝑚𝑜𝑑 4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([0]2, [15]2) = 4. 𝑓∗(𝑣𝑖𝑣𝑖 ′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([0]2, [63]2) = 6. 𝑓∗(𝑣𝑚𝑣𝑚 ′ ) =ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′ )]2)= hd([0]2, [15]2) = 4. Case (iv): if i ≡ 0(mod4); 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([15]2, [3]2) = 2 . 𝑓∗(𝑣𝑖𝑣𝑖 ′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([15]2, [48]2) = 6. 𝑓∗(𝑣𝑚𝑣𝑚 ′ ) =ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′ )]2)= hd([15]2, [3]2) = 2. From all the above cases, all the adjacent edges receive distinct Even labels. Hence the Path graph Pm admits even hamming distance labeling and the even hamming distance number is ηℎ𝑑 ′′ (𝑃𝑚 +) = 6. Algorithm 2.1.3. Even hamming Distance Labeling of Twig graph 𝑻𝑾(𝑷𝒎) Input: Vertices of 𝑇𝑊(𝑃𝑚) graph , 𝑚 ≥ 2 𝑉 ← {𝑣0 , 𝑣𝑚 , 𝑣𝑖 ′ , 𝑣𝑖 ′′/1 ≤ 𝑖 ≤ 𝑚 − 1} 𝑣0 ← 0; for 𝑖 = 1 𝑡𝑜 𝑚 do 𝑣𝑖 ← { 3 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 252 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 60 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 195 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) end for for 𝑖 = 1 𝑡𝑜 𝑚 − 1 do Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1922 https://internationalpubls.com 𝑣𝑖 ′ ← { 12 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 12 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) 𝑣𝑖 ′′ ← { 60 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 3 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 12 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) end for end procedure output: The labeled vertices of twig 𝑇𝑊(𝑃𝑚 )graph. Theorem 2.1.4. The Twig graph 𝑇𝑊(𝑃𝑚) is an even hamming distance labeled graph and the even hamming distance number is ηℎ𝑑 ′′ (𝑇𝑊(𝑃𝑚)) = 8, for any 𝑚 ≥ 2. Proof: Let us consider the twig graph 𝑇𝑊(𝑃𝑚) with vertex set 𝑉 = {𝑣0 , 𝑣𝑚 , 𝑣𝑖 ′ , 𝑣𝑖 ′′/1 ≤ 𝑖 ≤ 𝑚 − 1} and edge set 𝐸 = {{𝑣𝑖𝑣𝑖+1 / 0 ≤ 𝑖 ≤ 𝑚 − 1} ∪ {𝑣𝑖𝑣𝑖 ′ / 1 ≤ 𝑖 ≤ 𝑚 − 1} ∪ {𝑣𝑖𝑣𝑖 ′′ / 1 ≤ 𝑖 ≤ 𝑚 − 1}}. Define a function 𝑓: V → N ∪ {0} such that 𝑓(𝑢) ≠ 𝑓(𝑣) for any two adjacent vertices u and v as given in the above algorithm 2.1.3. Hence the adjacent vertices receive distinct labels. The edge labels are obtained as follows: 𝑓∗(𝑣0𝑣1) = ℎ𝑑([𝑓(𝑣0)]2, [𝑓(𝑣1)]2) = = hd([0]2, [3]2)=2. For 1 ≤ 𝑖 ≤ 𝑚 − 1, Case (i): if i ≡ 1(mod 4)or m ≡ 1(mod 4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([3]2, [252]2) = 8. 𝑓∗(𝑣𝑖𝑣𝑖 ′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([3]2, [12]2) = 4. 𝑓∗(𝑣𝑖𝑣𝑖 ′′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′′)]2) = hd([3]2, [60]2) = 6. 𝑓∗(vm−1vm) = hd([f(vm−1)]2, [f(vm)]2) = hd([195]2, [3]2) = 2. Case (ii): if i ≡ 2(mod4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([252]2, [60]2) = 2. 𝑓∗(𝑣𝑖𝑣𝑖 ′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd(2522, [12]2) = 4. 𝑓∗(𝑣𝑖𝑣𝑖 ′′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′′)]2) = hd([252]2, [0]2) = 6. 𝑓∗(vm−1vm) = hd([f(vm−1)]2, [f(vm)]2) = hd([3]2, [252]2) = 8. Case (iii): if i ≡ 3(mod 4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([60]2, [195]2) = 8. 𝑓∗(𝑣𝑖𝑣𝑖 ′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([60]2, [0]2) = 4. 𝑓∗(𝑣𝑖𝑣𝑖 ′′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′′)]2) = hd([60]2, [3]2) = 6. 𝑓∗(vm−1vm) = hd([f(vm−1)]2, [f(vm)]2) = hd([252]2, [60]2) = 2. Case (iv): if i ≡ 0(mod4) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1923 https://internationalpubls.com 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([195]2, [3]2) = 2 . 𝑓∗(𝑣𝑖𝑣𝑖 ′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([195]2, [0]2) = 4. 𝑓∗(𝑣𝑖𝑣𝑖 ′′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′′)]2) = hd([195]2, [12]2) = 6. 𝑓∗(vm−1vm) = hd([f(vm−1)]2, [f(vm)]2) = hd([60]2, [195]2) = 8. From all the above cases, all the adjacent edges receive distinct even labels. Hence the twig graph 𝑇𝑊(𝑃𝑚), admits even hamming distance labeling and the even hamming distance number is ηℎ𝑑 ′′ (𝑇𝑊(𝑃𝑚)) = 8, for any 𝑚 ≥ 2. Algorithm 2.1.5. Even Hamming Distance labeling of Centipede graph(𝒎, 𝟐). Input: Vertices of Centipede graph (𝑚, 2) graph, 𝑚 ≥ 1 𝑉 ← {𝑣𝑖 , 𝑣𝑖 ՚ , 𝑣𝑖 ՚՚/0 ≤ 𝑖 ≤ 𝑚} 𝑣0 ← 0; 𝑣0 ′ ← 15; 𝑣0 ′′ ← 63; 𝑣𝑚 ′ ← { 12 𝑖𝑓 𝑚 ≡ 1(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑚 ≡ 2(𝑚𝑜𝑑4) 15 𝑖𝑓 𝑚 ≡ 3(𝑚𝑜𝑑4) 3 𝑖𝑓 𝑚 ≡ 0(𝑚𝑜𝑑4) ; 𝑣𝑚 ′′ ← { 60 𝑖𝑓 𝑚 ≡ 1(𝑚𝑜𝑑4) 51 𝑖𝑓 𝑚 ≡ 2(𝑚𝑜𝑑4) 63 𝑖𝑓 𝑚 ≡ 3(𝑚𝑜𝑑4) 48 𝑖𝑓 𝑚 ≡ 0(𝑚𝑜𝑑4) for 𝑖 = 1 𝑡𝑜 𝑚 − 1 do 𝑣𝑖 ′ ← { 60 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 51 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 63 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 48 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) ; 𝑣𝑖 ′′ ← { 252 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 243 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 255 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 240 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) end for for 𝑖 = 1 𝑡𝑜 𝑚 do 𝑣𝑖 ← { 3 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 12 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 15 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) ; end for end procedure Output: The labeled vertices of Centipede (m,2) graph. Theorem: 2.1.6. The centipede graph (𝑚, 2) is an even hamming distance labeled graph and the even hamming distance number is ηℎ𝑑 ′′ (m, 2) = { 6, 𝑖𝑓 𝑚 = 1 8, 𝑖𝑓 𝑚 > 1 Proof: Let us consider the centipede graph (𝑚, 2) with vertex set 𝑉 = {{𝑣0,𝑣1,𝑣2,… , 𝑣𝑚} ∪ {𝑣0, ′ 𝑣1, ′ 𝑣2, ′ … , 𝑣𝑚 ′ } ∪ {𝑣0, ′′𝑣1, ′′𝑣2, ′′… , 𝑣𝑚 ′′}} and edge set 𝐸 = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1924 https://internationalpubls.com {{𝑣𝑖𝑣𝑖+1 / 0 ≤ 𝑖 ≤ 𝑚 − 1} ∪ {𝑣𝑖𝑣𝑖 ′/0 ≤ 𝑖 ≤ 𝑚} ∪ {𝑣𝑖𝑣𝑖 ′′/0 ≤ 𝑖 ≤ 𝑚}}. Define a function 𝑓: V → N ∪ {0} such that 𝑓(𝑢) ≠ 𝑓(𝑣) for any two adjacent vertices u and v as given in the above algorithm 2.1.5. hence the adjacent vertices receive distinct labels. The edge labels are obtained as follows: 𝑓∗(𝑣0𝑣0 ′) = ℎ𝑑([𝑓(𝑣0)]2, [𝑓(𝑣0 ′)]2) = hd([0]2, [15]2) = 4. 𝑓∗(𝑣0𝑣0 ′′) = ℎ𝑑([𝑓(𝑣0)]2, [𝑓(𝑣0 ′′)]2 = hd([0]2, [63]2) = 6. For 1 ≤ 𝑖 ≤ 𝑚 − 1, Case (i): if i ≡ 1(mod 4) or m ≡ 1(mod 4) 𝑓∗(𝑣𝑖𝑣𝑖 ′) = ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([3]2, [60]2) = 6. 𝑓∗(𝑣𝑖𝑣𝑖 ′′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′′)]2) = hd([3]2, [252]2) = 8. 𝑓∗(𝑣𝑚𝑣𝑚 ′ ) = ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′ )]2) = hd([3]2, [12]2) = 4. 𝑓∗(𝑣𝑚𝑣𝑚 ′′) = ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′′)]2) = hd([3]2, [60]2) = 6. Case (ii): if i ≡ 2(mod4) or m ≡ 2(mod 4) 𝑓∗(𝑣𝑖𝑣𝑖 ′) = ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([12]2, [51]2) = 6. 𝑓∗(𝑣𝑖𝑣𝑖 ′′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′′)]2) = hd([12]2, [243]2) = 8. 𝑓∗(𝑣𝑚𝑣𝑚 ′ ) = ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′ )]2) = hd([12]2, [0]2) = 2. 𝑓∗(𝑣𝑚𝑣𝑚 ′′) = ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′′)]2) = hd([12]2, [51]2) = 6. Case (iii): if i ≡ 3(mod 4) or m ≡ 3(mod 4) 𝑓∗(𝑣𝑖𝑣𝑖 ′) = ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([0]2, [63]2) = 6. 𝑓∗(𝑣𝑖𝑣𝑖 ′′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′′)]2) = hd([0]2, [255]2) = 8. 𝑓∗(𝑣𝑚𝑣𝑚 ′ ) = ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′ )]2) = hd([0]2, [15]2) = 4. 𝑓∗(𝑣𝑚𝑣𝑚 ′′) = ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′′)]2) = hd([0]2, [63]2) = 6. Case (iv): if i ≡ 0(mod4) or m ≡ 0(mod 4) 𝑓∗(𝑣𝑖𝑣𝑖 ′) = ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′)]2) = hd([15]2, [48]2) = 6. 𝑓∗(𝑣𝑖𝑣𝑖 ′′) =ℎ𝑑([𝑓(𝑣𝑖)]2, [𝑓(𝑣𝑖 ′′)]2) = hd([15]2, [240]2) = 8. 𝑓∗(𝑣𝑚𝑣𝑚 ′ ) = ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′ )]2) = hd([15]2, [3]2) = 2. 𝑓∗(𝑣𝑚𝑣𝑚 ′′) = ℎ𝑑([𝑓(𝑣𝑚)]2, [𝑓(𝑣𝑚 ′′)]2) = hd([15]2, [48]2) = 6. From all the above cases, all adjacent edges receive distinc event labels. Hence it is proved that the centipede graph (m,2) admits even hamming distance labeling and the even hamming distance number is ηℎ𝑑 ′′ (m, 2) = { 6, 𝑖𝑓 𝑚 = 1 8, 𝑖𝑓 𝑚 > 1 . Algorithm 2.1.7. Even Hamming Distance labeling of Comb Product of 𝑷𝒎 and 𝑷𝒓 : Input: Vertices of (𝑃𝑚 ⊳ 𝑃𝑟) graph, 𝑚, 𝑟 ≥ 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1925 https://internationalpubls.com 𝑉 ← {𝑣𝑖 , 𝑢𝑖 (𝑗)/0 ≤ 𝑖 ≤ 𝑚, 2 ≤ 𝑗 ≤ 𝑟 + 1, 𝑤ℎ𝑒𝑟𝑒 𝑣𝑖 = 𝑢𝑖 (1)} 𝑣0 = 𝑢0 (1) ← 0; 𝑢0 (2) ← 15; for 𝑖 = 1 𝑡𝑜 𝑚 do 𝑣𝑖 = 𝑢𝑖 (1) ← { 3 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 60 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 12 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 51 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) ;𝑢𝑖 (2) ← { 12 𝑖𝑓 𝑖 ≡ 1(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑖 ≡ 2(𝑚𝑜𝑑4) 3 𝑖𝑓 𝑖 ≡ 3(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑖 ≡ 0(𝑚𝑜𝑑4) end for for i = 0 to m do for j = 3 to r+1 do if i ≡ 1(𝑚𝑜𝑑2) do 𝑢𝑖 (𝑗) ← { 3 𝑖𝑓 𝑗 ≡ 1(𝑚𝑜𝑑4) 12 𝑖𝑓 𝑗 ≡ 2(𝑚𝑜𝑑4) 0 𝑖𝑓 𝑗 ≡ 3(𝑚𝑜𝑑4) 15 𝑖𝑓 𝑗 ≡ 0(𝑚𝑜𝑑4) else 𝑢𝑖 (𝑗) ← { 0 𝑖𝑓 𝑗 ≡ 1(𝑚𝑜𝑑4) 15 𝑖𝑓 𝑗 ≡ 2(𝑚𝑜𝑑4) 3 𝑖𝑓 𝑗 ≡ 3(𝑚𝑜𝑑4) 12 𝑖𝑓 𝑗 ≡ 0(𝑚𝑜𝑑4) end if end for end for end procedure output: The labeled vertices of Comb Product (𝑃𝑚 ⊳ 𝑃𝑟)graph. Theorem 2.1.8. The comb product of 𝑃𝑚 and 𝑃𝑟 graph (𝑃𝑚 ⊳ 𝑃𝑟)is an even hamming distance labeled graph and the even hamming distance number is ηℎ𝑑 ′′ (𝑃𝑚 ⊳ 𝑃𝑟) = { 4, 𝑖𝑓 𝑚 = 1 6, 𝑖𝑓 𝑚 > 1 . Proof: Let us consider the comb product of 𝑃𝑚 and 𝑃𝑟 graph (𝑃𝑚 ⊳ 𝑃𝑟) with vertex set 𝑉 = {𝑣𝑖 , 𝑢𝑖 (𝑗) /0 ≤ 𝑖 ≤ 𝑚, 2 ≤ 𝑗 ≤ 𝑟,𝑤ℎ𝑒𝑟𝑒 𝑣𝑖 = 𝑢𝑖 (1)} and edge se𝑡 𝐸 = {{𝑣𝑖𝑣𝑖+1 / 0 ≤ 𝑖 ≤ 𝑚 − 1} ∪ {𝑢𝑖 (𝑗) 𝑢𝑖+1 (𝑗+1) /0 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑟 − 1} }. Define a function 𝑓: V → N ∪ {0} such that 𝑓(𝑢) ≠ 𝑓(𝑣) for any two adjacent vertices u and v as given in the above algorithm 2.1.7. Hence the adjacent vertices Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1926 https://internationalpubls.com receive distinct labels. The edge labels are obtained as follows:𝑓∗(𝑢0 (1) 𝑢0 (2) ) = ℎ𝑑([𝑓(𝑢0 (1) )] 2 , [𝑓(𝑢0 (2) )] 2 ) = hd([0]2, [15]2) = 4. 𝑓∗(𝑢0 (2) 𝑢0 (3) ) = ℎ𝑑([𝑓(𝑢0 (2) )] 2 , [𝑓(𝑢0 (3) )] 2 ) = hd([15]2, [3]2) = 2. 𝑓∗(v0v1) = hd([f(v0)]2, [f(v1)]2) = hd([0]2, [3]2) = 2. For 1 ≤ 𝑖 ≤ 𝑚 − 1 Case (i): if 𝑖 ≡ 1(𝑚𝑜𝑑4) or 𝑚 ≡ 1(𝑚𝑜𝑑4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([3]2, [60]2) = 6. 𝑓∗(𝑢𝑖 (1) 𝑢𝑖 (2) ) = ℎ𝑑([𝑓(𝑢𝑖 (1) )] 2 , [𝑓(𝑢𝑖 (2) )] 2 ) = hd([3]2, [12]2) = 4. 𝑓∗(𝑢𝑖 (2) 𝑢𝑖 (3) ) = ℎ𝑑([𝑓(𝑢𝑖 (2) )] 2 , [𝑓(𝑢𝑖 (3) )] 2 ) = hd([12]2, [0]2) = 2. 𝑓∗(vm−1vm) = hd([f(vm−1)]2, [f(vm)]2) = hd([51]2, [3]2) = 2. Case (ii): if 𝑖 ≡ 2(𝑚𝑜𝑑4) or 𝑚 ≡ 2(𝑚𝑜𝑑4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([60]2, [12]2) = 2. 𝑓∗(𝑢𝑖 (1) 𝑢𝑖 (2) ) = ℎ𝑑([𝑓(𝑢𝑖 (1) )] 2 , [𝑓(𝑢𝑖 (2) )] 2 ) = hd([60]2, [0]2) = 4. 𝑓∗(𝑢𝑖 (2) 𝑢𝑖 (3) ) = ℎ𝑑([𝑓(𝑢𝑖 (2) )] 2 , [𝑓(𝑢𝑖 (3) )] 2 ) = hd([0]2, [3]2) = 2. 𝑓∗(vm−1vm) = hd([f(vm−1)]2, [f(vm)]2) = hd(32, [60]2) = 6. Case (iii): if 𝑖 ≡ 3(𝑚𝑜𝑑4) or 𝑚 ≡ 3(𝑚𝑜𝑑4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([12]2, [51]2) = 6. 𝑓∗(𝑢𝑖 (1) 𝑢𝑖 (2) ) = ℎ𝑑([𝑓(𝑢𝑖 (1) )] 2 , [𝑓(𝑢𝑖 (2) )] 2 ) = hd([12]2, [3]2) = 4. 𝑓∗(𝑢𝑖 (2) 𝑢𝑖 (3) ) = ℎ𝑑([𝑓(𝑢𝑖 (2) )] 2 , [𝑓(𝑢𝑖 (3) )] 2 ) = hd([3]2, [0]2) = 2. 𝑓∗(vm−1vm) = hd([f(vm−1)]2, [f(vm)]2) = hd([60]2, [12]2) = 2. Case (iv): if 𝑖 ≡ 0(𝑚𝑜𝑑4) or 𝑚 ≡ 0(𝑚𝑜𝑑4) 𝑓∗(vivi+1) = hd([f(vi)]2, [f(vi+1)]2) = hd([51]2, [3]2) = 2. 𝑓∗(𝑢𝑖 (1) 𝑢𝑖 (2) ) = ℎ𝑑([𝑓(𝑢𝑖 (1) )] 2 , [𝑓(𝑢𝑖 (2) )] 2 ) = hd(512, [0]2) = 4. 𝑓∗(𝑢𝑖 (2) 𝑢𝑖 (3) ) = ℎ𝑑([𝑓(𝑢𝑖 (2) )] 2 , [𝑓(𝑢𝑖 (3) )] 2 ) = hd([0]2, [3]2) = 2. 𝑓∗(vm−1vm) = hd([f(vm−1)]2, [f(vm)]2) = hd([12]2, [51]2) = 6. For 0 ≤ 𝑖 ≤ 𝑚; 3 ≤ 𝑗 ≤ 𝑟 if 𝑗 ≡ 1(𝑚𝑜𝑑4); 𝑓∗(𝑢𝑖 (𝑗) 𝑢𝑖 (𝑗+1) ) = ℎ𝑑([𝑓(𝑢𝑖 (𝑗) )] 2 , [𝑓(𝑢𝑖 (𝑗+1) )] 2 ) = hd([0]2, [15]2) = 4. if 𝑗 ≡ 2(𝑚𝑜𝑑4); 𝑓∗(𝑢𝑖 (𝑗) 𝑢𝑖 (𝑗+1) ) = ℎ𝑑([𝑓(𝑢𝑖 (𝑗) )] 2 , [𝑓(𝑢𝑖 (𝑗+1) )] 2 ) = hd([15]2, [3]2) = 2. if 𝑗 ≡ 3(𝑚𝑜𝑑4); 𝑓∗(𝑢𝑖 (𝑗) 𝑢𝑖 (𝑗+1) ) = ℎ𝑑([𝑓(𝑢𝑖 (𝑗) )] 2 , [𝑓(𝑢𝑖 (𝑗+1) )] 2 ) = hd([3]2, [12]2) = 4. if 𝑗 ≡ 0(𝑚𝑜𝑑4); 𝑓∗(𝑢𝑖 (𝑗) 𝑢𝑖 (𝑗+1) ) = ℎ𝑑([𝑓(𝑢𝑖 (𝑗) )] 2 , [𝑓(𝑢𝑖 (𝑗+1) )] 2 ) = hd([12]2, [0]2) = 2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1927 https://internationalpubls.com From all the above cases, all adjacent edges receive distinct even labels. Hence it is proved that the comb product of 𝑃𝑚 and 𝑃𝑟 graph (𝑃𝑚 ⊳ 𝑃𝑟) admits even hamming distance labeling and the even hamming distance number is ηℎ𝑑 ′′ (𝑃𝑚 ⊳ 𝑃𝑟) = { 4, 𝑖𝑓 𝑚 = 1 6, 𝑖𝑓 𝑚 > 1 . 3. Conclusion In this paper, the even hamming distance number of Comb graph 𝑃𝑚 +, Twig graph 𝑇𝑊(𝑃𝑚), Centipede graph(𝑚, 2), and comb product of 𝑃𝑚 and 𝑃𝑟 graph(𝑃𝑚 ⊳ 𝑃𝑟) were obtained. References [1].Durai Baskar.A and Manivannan.P, F-Heronian Mean labeling of graphs, International Journal of Pure and Applied Mathematics, Volume 117(5), 2017, 55-62. [2].Esakkiammal.E. Thirusangu.K and Seethalakshmi.S, d-lucky labeling of arbitrary super subdivision of some graphs, International Journal of Pure and Applied Mathematics, Volume 113(7), 2017, 93-101. [3] Gallian J.A. A Dynamic Survey of Graph Labeling, The Electronic Journal Of Combinatorics. 2019. DS6. [4] Nikitha Prakash and Supriya Rajedran, Neighbourhood prime labelling on some path related graphs, International Journal of Pure and Applied Mathematics,Volume 118(20), 2018, 1893- 1901. [5] Seethalakshmi.S, Thirusangu.K, Esakkiammal.E, Hamming Distance labeling of Certain graphs, Journal of Tianjin University Science and Technology, 2021 54(10), 106-113, DOI:10.17605/OSF.IO/FMA2Q. [6] Sunoj B.S, Mathew Varkey T K, Square Difference Prime Labeling for Some Tree Graphs, IJEDR, 2017, Vol.5(4), ISSN :2321-9939.