INFOKARA RESEARCH ISSN NO: 1021-9056
VERTEX POLYNOMIAL OF LADDER GRAPHS
1V. Jeba Rani, 2S. Sunder Raj and 3T. Shyla Isaac Mary, 1Register Number.12578 Research Scholar. Nesamony Memorial Christian College, Marthandam – 629165, India. [email protected] 2Associate Professor, Department of Mathematics Vivekananda College, Agasteeswaram – 629203 India. 3 Assistant Professor,Department of Mathematics Nesamony Memorial Christian College, Marthandam – 629165, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, India
ABSTRACT Let G = (V, E) be a graph. The vertex polynomial of the graph G = (V, E) is
∆( ) defined as Vp (G, x) = ∑ 푣 푥 where ∆ (G) = max {d (v) / v휖V) and 푣 is the number of vertices of degree i. In this paper we derived the vertex polynomial for Ladder graph, Circular Ladder graph, Mobius Ladder graph, Triangular Ladder graph, Diagonal Ladder graph, Step Ladder graph, and Double Sided Step Ladder graph. Keywords: Ladder graph, Circular Ladder graph, Mobius Ladder graph, Triangular Ladder graph, Diagonal Ladder graph, Step Ladder graph, Double Sided Step Ladder graph, Vertex Polynomial.
1. INTRODUCTION In a graph 퐺 = (푉, 퐸) we mean a finite undirected, non-trivial graph without loops and multiple edges. The vertex set is denoted by V and the edge set by E, for 푣 ∈ 푉푑(푣) is the number of edges incident with v, the maximum degree of G is defined as ∆(G) = max {푑(푣)/푣 ∈∈ 푉} for terms not defined here, we refer to Frank Harary [1]. A cubic graph is a graph in which all the vertices have degree 3.
Definition: 1:1The Ladder graph Ln is a Planer undirected graph with 2n vertices and 3n-2 edges. The Ladder graph can be obtained as the Cartesian product of two path graph one of
which has only one edge Ln,1=Pn x P2. It has 2n vertices and 3n-2 edges.
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Definition:1:2 The Circular Ladder graph CLn is visualized as two concentric n-cycles in which each of n pairs of corresponding vertices is joined by an edge. It has 2n vertices and 3n edges.
Definition: 1:3 The Mobius Ladder graph MLn is a simple cubic graph on 2n vertices. The graph
MLncontains ‘n’ diagonals and 2n outside edges (n≥2)
Definition:1:4 A Triangular Ladder graph TLn (n≥2) is a graph obtained from a Ladder graph Ln
= Pn x P2by adding the edges uivi+1for 1≤i≤ n-1such graph has 2n vertices with 4n-3 edges.
Definition: 1:5 The Diagonal Ladder graph DLn (n≥2) is a graph obtained from a Ladder graph
by adding the edges uivi+1 and ui +1vi for1 ≤ i ≤ n-1such graph has 2n vertices and 5n-4 edges.
2. MAIN RESULTS
Theorem: 2:1Let Lnbe a Ladder graph with 2n vertices (n ≥ 2). The vertex polynomial of Ln is 2 3 Vp (Ln,x) = 4x + 2 (n-2) x , n ≥2
Proof:Let Ln be a Ladder graph with 2n vertices (n≥2) and 3n-2 edges. In the Ladder graph Ln have four corner points with degree 2 and the remaining vertices have degree 3 it gives the result. Example: 2:2Take n=3 in the above theorem. We have the graph.
푣 푣 푣
푣 푣 푣 퐺 Figure 2.1
2 3 Here Vp(Ln,x) = 4x + 2x
Theorem: 2:3 Let Ln be a Ladder graph with 2n vertices (n≥2) and
휉 = Ln∪Ln∪Ln∪… ∪Ln(m times). Then the vertex polynomial is 2 3 Vp(ξ,x) = m (4x ) + 2m(n-2)x , n≥2, m≥1
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Proof:In this theorem we have to consider m copies of Ln, here the number of vertices of Ln is
increased by m copies but degree of each vertex remains unchanged. Hence each co-efficient of
the vertex polynomial of G multiplied by m it gives the result.
Theorem:2:4 Let CLnbe a circular Ladder graph with 2n vertices (n≥ 2). The vertex polynomial
3 of CLn is Vp (CLn, x) = 2nx , (n≥ 2).
Proof:Let CLn be a circular Ladder graph with 2n vertices (n≥2) and 3n edges. In a Circular
Ladder graph each vertices of inner circle have degree 3 because every vertex connected to two
other vertices in inner circle and one vertex in outer circle. In outer circle each vertex connected
to two other vertices in outer circle and one vertex in inner circle. So every vertex has degree 3
it gives the result.
Theorem:2:5 Let CLn be a Circular Ladder graph with 2n vertices (n ≥ 2) and ξ =
3 CLn∪CLn∪CLn∪… ∪CLn (m times). Then the vertex polynomial is Vp (ξ,x) = 2 mnx , n ≥ 2, m
≥ 1.
Theorem:2:6 Let MLn be a Mobius Ladder graph with 2n vertices (n≥2). The vertex
3 Polynomial of MLn is Vp (MLn,x) = 2nx , (n≥2).
Proof:Let MLn be a Mobius Ladder graph with 2n vertices (n≥2). By definition of Mobius
Ladder graph is a simple cubic graph. So every vertex has degree 3 it gives the result.
Result: 2:7 The circular Ladder graph and Mobius Ladder graph have the same vertex
polynomial.
Theorem:2:8 Let TLnbe a Triangular Ladder graph with 2n vertices (n≥2). The vertex
2 3 4 polynomial of TLn is Vp (TLn,x) = 2x +2x +2(n-2) x , (n≥2).
Proof: Let TLn be a Triangular Ladder graph with 2n vertices (n≥2). A Triangular Ladder graph
have four corner vertices, the two of the opposite corner vertices have degree 2 and rest of the
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corner vertices have degree 3 because the vertices ui and vi+1 are joined by edge. The remaining
inner vertices have degree four it gives the result.
Example:2:9 Take 푛 =4 in the above theorem .We have the graph
푢 푢 푢 푢
푣 푣 푣 푣 G Figure 2.2
2 3 4 Here Vp (TL4, x) = 2x +2x +4 x
Theorem:2:10 Let TLn be a Triangular Ladder graph with 2n vertices (n ≥ 2) and ξ = 2 3 TLn∪TLn∪TLn∪…∪TLn (m times). Then the vertex polynomial is Vp (ξ,x) =2mx +2mx +2m(n- 2) x4, n ≥ 2, m ≥ 1
Theorem:2:11 Let DLn be a Diagonal Ladder graph with 2n vertices (n≥2). The vertex 3 5 polynomial of DLn is Vp(DLn,x) = 4x +2(n-2)x , n≥2
Proof:Let DLn be a Diagonal Ladder graph with 2n vertices (n≥2). In a Diagonal Ladder graph
have four corner vertices each of degree 3 because in a Diagonal Ladder graph the vertices ui,vi+1
and ui+1, vi are joined by edges. The remaining vertices have degree 5 it gives the result.
Theorem: 2:12Let DLn be a Triangular Ladder graph with 2n vertices (n ≥ 2) and
ξ = DLn∪DLn∪DLn∪… ∪DLn(m times). Then the vertex polynomial is 3 5 Vp (ξ,x) = 4mx +2m(n-2)x , n ≥ 2, m ≥ 1
Definition: 2:13 [6] Let Pn be a path of n vertices denoted by (1,1), (1,2) …..,(1,n) and n-1 edges
denoted by e1,e2,…………en-1 where ei is the edge joining the vertices (1,i) and (1, i+1)on each
edge ei, i=1,2,….n-1 we erect a Ladder with n-(i-1) steps including the edge ei.The graph
obtained is called a Step Ladder graph and is denoted by S(Ln) where ‘n’ denotes the number of
vertices in the base. It has vertices and n(n+1)-2 edges.
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Theorem: 2:14 Let S(Ln) be a Step Ladder graph with n vertices (n≥ 2). The vertex polynomial 2 3 4 of S(Ln) is Vp (SLn,x) = (n+2) x +(2n-4) x + ( ) x , n ≥ 2 Proof:Let S(Ln) be a step Ladder graph with n vertices (n≥2). The graph S(Ln) has (n+2) corner vertices each of degree 2. The vertices in boundary lines have degree 3 except the corner
vertices. The remaining ( ) inner vertices have degree 4 because each of the inner vertices join four adjacent vertices it gives the result.
Example:2:15 Take n=6 in the above theorem .we have the graph
푣 푣
푣 푣 푣
푣 푣 푣 푣
푣 푣 푣 푣 푣
푣 푣 푣 푣 푣 푣
푣 푣 푣 푣 푣 푣
퐺 Figure 2.3
2 3 4 Here Vp (SL6, x) = 8x +8x +10x
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Theorem: 2:16 Let S(Ln) be a Step Ladder graph with n vertices (n ≥ 2) andξ = SLn∪SLn∪ 2 SLn∪… ∪SLn(m times). Then the vertex polynomial of S(Ln) is Vp (ξ, x) =m(n+2) x +m(2n-4)
x3+푚 ( ) x4, n ≥ 2, m ≥1.
Definition: 2:17 [6] Let P2n be a path of length 2n-1 with 2n vertices (1,1), (1,2),……….,(1,2n)
with 2n-1 edges e1, e2,……..,e2n-1 where ei is the edge joining the vertices (1,i) and (1,i+1) on
each edge ei for i=1,2,…………n.We erect a Ladder with i+1 steps including the edge ei and on
each edge ei for i =n+1, n+2,…..,2n-1 we erect a Ladder with 2n+1-i steps including the edge ei.
The Double Sided Step Ladder graph 2S (L2n) has vertices denoted by (1,1), (1,2),….,(1,2n), (2,1), (2,2),….,(2,2n), (3,2), (3,3),….,(3,2n-1), (4,3),(4,4),….,(4,2n-2),….,(n+1,n), (n+1,n+1).In the ordered pair (i,j) i denotes the row number (counted from bottom to top) and j denotes the column number (from left to right) in which the vertex occurs. It has n2+3n vertices and 2n2+3n- 1 edges.
Theorem:2:18 Let 2S(L2n) be a Double Sided Step Ladder graph with 2n vertices (n≥ 1). The 2 3 2 4 vertex polynomial of 2S(L2n) is Vp(2S(L2n),x) = (2n+2) x +(2n-2) x +(n -n) x , n≥1
Proof:Let 2S(L2n) be a Double Sided Step Ladder graph with 2n vertices (n≥1). The graph
2S(L2n) has 2n+2 corner vertices and each of have degree 2. The total number of vertices in base of the graph except the corner vertices are 2n-2 each of have degree 3 and the remaining n2-n vertices have degree 4 it gives the result.
Example: 2:19 Take n=3 in the above theorem .we have the graph
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푣 푣
푣 푣 푣 푣
푣 푣 푣 푣 푣 푣
푣 푣 푣 푣 푣 푣
퐺 Figure 2.4
2 3 4 Here V(2S(L6),x) = 8x + 4x + 6x
Theorem: 2:20Let 2S(L2n) be a Double Sided Step Ladder graph with 2n vertices (n ≥ 1) and ξ =
2S(L2n)∪2S(L2n)∪2S(L2n)∪… ∪2S(L2n) (m times). Then the vertex polynomial is Vp(ξ, x) = m(2n+2)x2+m(2n-2)x3+m(n2-n)x4, n ≥ 1 ,m ≥ 1
REFERENCE [1] Frank Harary, 1872, “Graph Theory”, Addition – Wesly publishing company. [2] E.Ebin Raja Merly, A.M.Anto, “Vertex Polynomial for the Splitting graph of Comb and Crown, International journal of Emerging Technologies in Engineering Research (IJETER) volume 4, Issue 10, October (2016) [3] J. Devaraj, E.Sukumaran “On vertex polynomial” International J. of Math.sci engg Appls (IJMESA) vol.6 No.1 (January 2012) PP.371-380 [4] Hosoya.H and Harary.F “On the Matching properties of Three Fence Graphs”. Journal of Mathematical chemistry. 12, 211-218,1993 [5] John P. Mcsoreley “Counting structures in the Mobius Ladder” published in discrete mathematics 184(1-3), 137-164
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[6] M.A. Seoud, Shakir M. Salman, “On Difference Cordial Graphs,” Mathematica Aeterna, Vol.5, 2015, no.1, 105-124 [7] E. Ebin Raja Merly and A.M.Anto, “Adjacent vertex sum polynomial of factographs,” Journal of Global Research in Mathematical Archives, Volume 2, No.11, November 2014, 25- 29.
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