Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 424 https://internationalpubls.com On Irregular Coloring of Middle Graph of Snake Graphs Dr. G. Jothilakshmi.1, Dr. K. Rajam 2, Dr. S. Manimekalai 3, Dr. U. S. Monolisa4, K. Nandhini5 1,5 Assistant Professor, Department of Mathematics, RVS College of Arts and Science (Affiliated to Bharathiar University) Coimbatore, India. Email: jothikrishna11@gmail.com 2, Assistant Professor (SS), Department of Mathematics, Dr. NGP Institute of Technology (Affiliated to Anna University) Coimbatore, India, Email: rajee.mat@gmail.com 3, Associate Professor, Department of Mathematics, Dr.N.G.P Arts and Science College (Affiliated to Bharathiar University) Coimbatore, India Email:Manimekalai@drngpasc.ac.in 4 Assistant Professor, Department of Mathematics, Dr. Mahalingam College of Engineering and Technology (Affiliated to Anna University) Coimbatore, India Email: monolisamat@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: An irregular coloring is a proper coloring in which distinct vertices have different color codes. In this paper we obtain the irregular chromatic number of middle graph of certain snake graphs of snake graph families. Keywords: Double triangular snake graph, Diamond snake graph, Graph coloring, Irregular chromatic number, Irregular coloring , Middle graph ,Triangular snake graph Classification number: 05C15, 05C76 1. Introduction Let G be a finite, undirected graph [1] with no loops and multiple edges. The graph G has the vertex set V (G) and the edge set E (G). Graph coloring [5] is coloring of G such that no two adjacent vertices share the same color. A proper coloring c is an irregular coloring [1] if no two like-colored vertices have the same color code. i.e., for every pair of vertices u and w; code(u) ≠code(w) whenever c(u) = c(w). Thus, an irregular coloring distinguishes each vertex from each of other vertex by its color or by its color code. Mary Radcliffe and Ping Zhang[8,9] established sharp upper and lower bounds for the irregular chromatic number of a disconnected graph in terms of the irregular chromatic numbers of its components. A.Rohini and M.Venkatachalam [11] discussed the irregular coloring of wheel related graphs. Further this paper exhibits the irregular coloring of snake graph families. The central graph C(G) is obtained by subdividing each edge of G exactly once and joining all the non-adjacent vertices of G. The vertex set of middle graph M(G) is V(G)∪E(G) in which two elements are adjacent in M(G) if the following conditions hold. (i) x,y ∈ E(G) and x, y are adjacent in G. (ii) x ∈V(G), y ∈ E(G) and y is incident on x in G. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 425 https://internationalpubls.com A triangular snake graph Tn [3] is obtained from a path v1, v2, v3,…….vn+1 ,u1, u2 …. un .It has 2n+1 vertices and 3n edges, where n is the number of blocks in the triangular snake and a double triangular snake graph D(Tn) [3] is obtained from a path v1, v2, v3,…….vn+1 by joining vi and vi+1 to a new vertex wi to another new vertex ui . A diamond snake graph Dn [3] is obtained from a path v1, v2, v3,…….vn+1 by joining vi and vi+1 to a new vertex wi to another new vertex ui 2. Structural properties of middle graph of M (𝐓𝐧), M [D(Tn)] and M[Dn] • Number of vertices in M (Tn), p= 5n+1 • Number of vertices in M[D(Tn)] ,p=8n+1 • Number of vertices in M[Dn] ,p=7n+1 • Maximum degree in M (Tn) and M[Dn], ∆=6 • Maximum degree in M[D(Tn)], ∆=7 • Minimum degree in M(Tn),M[Dn], δ =2 • Minimum degree in M[D(Tn)], δ = 3 3. IRREGULAR COLORING OF M (𝐓𝐧), M [D(Tn)] and M[Dn] Theorem 3.1 : For M(𝑇n), 𝜒𝑖𝑟[ [M (Tn)] = 2𝑛 + 2 , n≥2 Proof: Now by definition of middle graph, each edge of the triangular snake graph Tn is sub-divided by a new vertex. Assume that each edge (vi,vi+1) and the line joining vi and vi+1 to a vertex ui ,i=1,2,3,….n are sub- divided by the vertices wi, ejj and ejj+1, j=1,2,3,….n respectively. Assign the color c1 to vi, ui and c2, c3….c2n+1 to e1,e 2……e2n Assign the color cn+3, cn+4….. c2n+2 to e2n+1,e2n+2……e3n To prove (2n+2)- coloring is an irregular coloring of M (Tn), since deg(ui) ≠ deg(ei),it shows that code(ui) ≠ code (ei) Since each ui ’s are adjacent to ei’s but vi ’s are not adjacent to ui’s. Hence code (ei) ≠code(vi) .Thus 𝜒𝑖𝑟[M (Tn)] ≤ 2𝑛 + 2 By the definition of middle graph{v,ei}induces a clique of order 2n+2 in M(Tn) [9] It follows that,𝜒𝑖𝑟[M (Tn)] ≥ 𝜒[M (Tn)] = 2𝑛 + 2. Hence 𝜒𝑖𝑟[ [M (Tn)] = 2𝑛 + 2 , n≥2 By the definition of middle graph{v,ei}induces a clique of order 2n+2 in M(Tn) [9] It follows that,𝜒𝑖𝑟[M (Tn)] ≥ 𝜒[M (Tn)] = 2𝑛 + 2. Hence 𝜒𝑖𝑟[ [M (Tn)] = 2𝑛 + 2 , n≥2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 426 https://internationalpubls.com Figure 1: Middle graph of Triangular snake graph M(T2) Theorem 3.2 : For [M(D (Tn))] ,the irregular - chromatic number is 6. (𝑖. 𝑒) 𝜒𝑖𝑟 [M(D (Tn))] = 3𝑛 + 3 n ≥ 2 Proof: Let {v1,v2………vn+1 , u1,u2………un ,w1,w2……wn} be the vertices of the double triangular snake graph D(Tn). Assume that each edge (vi,vi+1) (ui , vj )and (vj , wi) , i=1,2,3,….n and j=1,2,3…n+1 is sub-divided by the vertices xi, eij and fij for , i=1,2,3,….n and j=1,2,3…n+1 respectively.[9] Color the vertices v1,v2………vn+1 ,u1,u2………un and w1,w2……wn with c1. Color the elements e1,e 2……e2n with c2, c3…. c2n+1 and e2n+1,e2n+2……e4n with c5, c6…..c2n+4 Atlast, Color the sub-divided vertices vi,i+1,vi+1,i+2….vn,i+n with c2n+4, c2n+5…. c3n+3 Hence 𝜒𝑖𝑟 [M (D(Tn)] = 3𝑛 + 3 , n≥2 Figure 2: Middle graph of Double Triangular snake graph M(D(T2)) Theorem 3.3: For [M (𝐷n)], the irregular -chromatic number is 2n+3 (𝑖. 𝑒 )𝜒𝑖𝑟 [M (𝐷n)] =2n+3, n≥2 Proof: Color the vertices {v1,v2………vn+1 , u1,u2………un ,w1,w2,….wn }with c1. Color the elements e11,e 12……e2n with c2, c3. …. c2n+1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 427 https://internationalpubls.com Color the elements e2n+1,e2n+2……e4n with c4, c5…..c2n+3. To prove (2n+3)- coloring is an irregular coloring of M (Dn), since deg(ui) ≠ deg(ei),it shows that code(ui) ≠ code (ei) Since each ui ’s are adjacent to ei’s but vi ’s are not adjacent to ui’s. Hence code (ei) ≠code(vi) . Thus 𝜒𝑖𝑟[M (Dn)] ≤ 2𝑛 + 3 By the definition of middle graph,{v,ei} induces a clique of order 2n+3 in M(Dn)[9] It follows that,𝜒𝑖𝑟[M (Dn)] ≥ 𝜒[M (Dn)] = 2𝑛 + 3 Figure 3: Middle graph of Double Triangular snake graph M(D2). 5. Conclusion An irregular coloring play an important role in clustering , automatic reading system and distributed system.The investigation of similar results for different graphs as well in the context of various graph coloring problems is an open area of research. References [1] J. A. Bondy and U. S. R. Murty, Graph theory with Applications, London Macmillan [2] Frank Harary, Narosa Publishing Home, (2001). [3] M.S.Franklin Thamilselvi, Harmonious coloring of central graph of certain graphs, Applied Mathematical sciences,5 pp.569-578(2015) [4] G.Jothilakshmi, On harmonious coloring of C(Hn) and C(Gn), Journal of Global Research in Mathematical Archives ,Volume 2, No. 5, (May 2014) [5] G.Jothilakshmi K.Rajam and A.Sathyakala,A Study on an Irregular Colouring of Sunlet and Pangraph Families, Indian Journal of Natural Sciences, Vol.15 , Issue 87 ,( Dec 2024) [6] Mary.U, G.Jothilakshmi, On harmonious coloring of M(Sn) and M(D3 m), International journal of computer Applications,vol 4,pp.239-244(2014) [7] Mary.U and G.Jothilakshmi, A study on harmonious coloring of Middle graph of certain snake graphs, Journal Of Applied Science And Computations, vol 6,pp.30-37(2019) [8] Radcliffe, M. and Zhang, P., Irregular coloring of graphs, Bull. Inst. Combin. Appl.,49, (2007), 41-59. [9] A.Rohini and M.Venkatachalam , On irregular coloring of wheel related graphs, International conference on current scenario in Pure and Applied Mathematics ,1462- 1472(2019).