Different ways to describe a permutation, as a sequence of integers, or a product of Coxeter gene... more Different ways to describe a permutation, as a sequence of integers, or a product of Coxeter generators, or a tree, give different choices to define a simple permutation. We recollect few of them, define new types of simple permutations, and analyze their interconnections and some asymptotic and geometrical properties of these classes.
Tomescu and Imran [Graphs and Combinatorics, 2010] proposed the following conjecture : for every ... more Tomescu and Imran [Graphs and Combinatorics, 2010] proposed the following conjecture : for every connected graph of order n the number of resolving pairs (i.e., pairs of vertices of G having distinct distances to all vertices of G) is bounded above by ⌊n 2 /4⌋ and solved into affirmative this assertion for graphs with diameter two. In this paper the conjecture is verified for bipartite graphs, graphs of order n and diameter n − 2 and for a subclass of graphs of diameter three. It is also shown that for every integers n,k such that n ≥ 3 and 2 ≤ k ≤ n − 1 there is a graph of order n and diameter k having ⌊n 2 /4⌋ resolving pairs.
Topological indices are numerical numbers assigned to the graph/structure and are useful to predi... more Topological indices are numerical numbers assigned to the graph/structure and are useful to predict certain physical/chemical properties. In this paper, we give explicit expressions of novel Banhatti indices, namely, first K Banhatti index B 1 G , second K Banhatti index B 2 G , first K hyper-Banhatti index HB 1 G , second K hyper-Banhatti index HB 2 G , and K Banhatti harmonic index H b G for hyaluronic acid curcumin and hydroxychloroquine. The multiplicative version of these indices is also computed for these structures.
Let G = (V, E) be a finite, simple and undirected graph having v = |V (G)| and e = |E(G)|. A grap... more Let G = (V, E) be a finite, simple and undirected graph having v = |V (G)| and e = |E(G)|. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2, . . . , 2q−1} such that, when each edge xy is assigned the label |f(x)−f(y)|, the resulting edge labels are {1, 3, 5, . . . , 2q−1}. Motivated by the work of Z. Gao [6], we have defined odd graceful labeling for some other union of graphs. In this paper we formulate odd-graceful labeling for disjoint unions of graphs consisting of generalized combs, ladder, star, bistar, caterpillar and path.
Kragujevac tree is indicated by K ; K ∈ K g q = s 2 t + 1 + 1 , s with order and size s 2 t + 1 +... more Kragujevac tree is indicated by K ; K ∈ K g q = s 2 t + 1 + 1 , s with order and size s 2 t + 1 + 1 and s 2 t + 1 , respectively. In this paper, we have a look at certain topological features of the total graph and line graph of the total graph of the considered tree, i.e ., Kragujevac tree, by computing different topological indices and polynomials.
An edge-magic total labeling of an (n,m)-graph G = (V,E) is a one to one map λ from V(G) ∪ E(G) o... more An edge-magic total labeling of an (n,m)-graph G = (V,E) is a one to one map λ from V(G) ∪ E(G) onto the integers {1,2,…,n + m} with the property that there exists an integer constant c such that λ(x) + λ(y) + λ(xy) = c for any xy ∈ E(G). It is called super edge-magic total labeling if λ (V(G)) = {1,2,…,n}. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edge-magic total labeling, called super edge-magic deficiency of a graph G, is denoted by μs(G) [4]. If such vertices do not exist, then deficiency of G will be + ∞. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].
A super edge-magic total (SEMT) labeling of a graph ℘(V, E) is a one-one map ϒ from V(℘)∪E(℘) ont... more A super edge-magic total (SEMT) labeling of a graph ℘(V, E) is a one-one map ϒ from V(℘)∪E(℘) onto {1, 2,…,|V (℘)∪E(℘) |} such that ∃ a constant “a” satisfying ϒ(υ) + ϒ(υν) + ϒ(ν) = a, for each edge υν ∈E(℘), moreover all vertices must receive the smallest labels. The super edge-magic total (SEMT) strength, sm(℘), of a graph ℘ is the minimum of all magic constants a(ϒ), where the minimum runs over all the SEMT labelings of ℘. This minimum is defined only if the graph has at least one such SEMT labeling. Furthermore, the super edge-magic total (SEMT) deficiency for a graph ℘, signified as $\mu_{s}(\wp)$ is the least non-negative integer n so that ℘∪nK1 has a SEMT labeling or +∞ if such n does not exist. In this paper, we will formulate the results on SEMT labeling and deficiency of fork, H -tree and disjoint union of fork with star, bistar and path. Moreover, we will evaluate the SEMT strength for trees.
Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd... more Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2,. .. , 2q− 1} such that, when each edge xy is assigned the label |f (x)− f (y)| , the resulting edge labels are {1, 3, 5,. .. , 2q− 1} and f is called an odd graceful labeling of G. Motivated by the work of Z. Gao [6] in which he studied the odd graceful labeling of union of any number of paths and union of any number of stars, we have determined odd graceful labeling for some other union of graphs. In this paper we formulate odd-graceful labeling for disjoint unions of graphs consisting of generalized combs, stars, bistars and paths.
Let G be a connected graph and d(x, y) be the distance between the vertices x and y. A subset of ... more Let G be a connected graph and d(x, y) be the distance between the vertices x and y. A subset of vertices W = {w1,w2, · · · ,wk} is called a resolving set for G if for every two distinct vertices x, y ∈ V (G), there is a vertex wi ∈ W such that d(x,wi) 6= d(y,wi). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension dim(G). Let F be a family of connected graphs Gn : F = (Gn)n�1 depending on n as follows: the order |V (G)| = '(n) and limn!1'(n) = ∞. If there exists a constant C > 0 such that dim(G) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension; otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), then F is called a family with constant metric dimension. The metric dimension of some classes of plane graphs has been determined in [3], [4], [5], [10], [12], [15] and [22], while...
Let G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of v... more Let G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of vertices W={w 1 ,w 2 ,⋯,w k } is called a resolving set for G if for every two distinct vertices x,y∈V(G), there is a vertex w i ∈W such that d(x,w i )≠d(y,w i ). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension dim(G). Let ℱ be a family of connected graphs G n :ℱ=(G n ) n≥1 depending on n as follows: the order |V(G)|=φ(n) and lim n→∞ φ(n)=∞. If there exists a constant C>0 such that dim(G)≤C for every n≥1 then we shall say that ℱ has bounded metric dimension; otherwise ℱ has unbounded metric dimension. If all graphs in ℱ have the same metric dimension (which does not depend on n), then ℱ is called a family with constant metric dimension. The metric dimension of some classes of plane graphs has been determined in several papers, while metric dimension of some classes o convex polyt...
We extend results about critically k-colorable graphs to choosability and paintability (list colo... more We extend results about critically k-colorable graphs to choosability and paintability (list colorability and on-line list colorability). Using a strong version of Brooks' Theorem, we generalize Gallai's Theorem about the structure of the low-degree subgraph of critically k-colorable graphs, and introduce a more adequate lowest-degree subgraph. We prove lower bounds for the edge density of critical graphs, and generalize Heawood's Map-Coloring Theorem about graphs on higher surfaces to paintability. We also show that on a fixed given surface, there are only finitely many critically k-paintable/k-choosable/ k-colorable graphs, if k ≥ 6. In this situation, we can determine in polynomial time k-paintability, k-choosability and k-colorability, by giving a polynomial time coloring strategy for ''Mrs. Correct''. Our generalizations of k-choosability theorems also concern the treatment of non-constant list sizes (non-constant k). Finally, we use a Ramsey-type lemma to deduce all 2-paintable, 2-choosable, critically 3-paintable and critically 3-choosable graphs, with respect to vertex deletion and to edge deletion.
Abstract: Different ways to describe a permutation, as a sequence of integers, or a product of Co... more Abstract: Different ways to describe a permutation, as a sequence of integers, or a product of Coxeter generators, or a tree, give different choices to define a simple permutation. We recollect few of them, define new types of simple permutations, and analyze their ...
Abstract: The paper contains enumerative combinatorics for positive braids, square free braids, a... more Abstract: The paper contains enumerative combinatorics for positive braids, square free braids, and simple braids, emphasizing connections with classical Fibonacci sequence. The simple subgraph of the Cayley graph of the braid group is analyzed in the final part.
Nanofluids are a very productive etymology of intensifying the process of heat and mass transport... more Nanofluids are a very productive etymology of intensifying the process of heat and mass transport systems linked with the industrial and thermal engineering systems. Nanomaterials have effective thermal properties and various applications in our daily life like in heat transfer, electronic cooling systems, energy production and biomedicine and also in the food industry. Keeping the entire motivating potential ramifications of nanoparticles in mind, this work is visualized in the mathematical model developed to show the heat and mass transport behavior of swimming motile organisms in the existence of the magnetic field, heat conduction source, thermal radiation, chemical processes and viscous dissipation. The flow of mass and heat transport under consideration is governed by nonlinear partial differential equations (PDEs) transformed into ordinary differential equations (ODEs) by implementing an eminent method called similarity transform and then numerical results obtained through MA...
Kragujevac tree is indicated by K; K ∈ Kg q�s(2t+1)+1,s with order and size s(2t + 1) + 1 and s(2... more Kragujevac tree is indicated by K; K ∈ Kg q�s(2t+1)+1,s with order and size s(2t + 1) + 1 and s(2t + 1), respectively. In this paper, we have a look at certain topological features of the total graph and line graph of the total graph of the considered tree, i.e., Kragujevac tree, by computing different topological indices and polynomials.
UPB Scientific Bulletin, Series A: Applied Mathematics and Physics , 2017
Tomescu and Imran [Graphs and Combinatorics, 2010] proposed the following conjecture : for every ... more Tomescu and Imran [Graphs and Combinatorics, 2010] proposed the following conjecture : for every connected graph of order n the number of resolving pairs (i.e., pairs of vertices of G having distinct distances to all vertices of G) is bounded above by ⌊n 2 /4⌋ and solved into affirmative this assertion for graphs with diameter two. In this paper the conjecture is verified for bipartite graphs, graphs of order n and diameter n − 2 and for a subclass of graphs of diameter three. It is also shown that for every integers n,k such that n ≥ 3 and 2 ≤ k ≤ n − 1 there is a graph of order n and diameter k having ⌊n 2 /4⌋ resolving pairs.
It is called super edge-magic total labeling if .V .G// D f1; 2; : : : ; ng. Furthermore, if G ha... more It is called super edge-magic total labeling if .V .G// D f1; 2; : : : ; ng. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edgemagic total labeling, called super edge-magic deficiency of a graph G, is denoted by s .G/ [4]. If such vertices do not exist, then deficiency of G will be C1. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].
Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd... more Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2,. .. , 2q − 1} such that, when each edge xy is assigned the label |f (x) − f (y)|, the resulting edge labels are {1, 3, 5,. .. , 2q − 1}. Motivated by the work of Z. Gao [6], we have defined odd graceful labeling for some other union of graphs. In this paper we formulate odd-graceful labeling for disjoint unions of graphs consisting of generalized combs, ladders, stars, bistars, caterpillars and paths .
Different ways to describe a permutation, as a sequence of integers, or a product of Coxeter gene... more Different ways to describe a permutation, as a sequence of integers, or a product of Coxeter generators, or a tree, give different choices to define a simple permutation. We recollect few of them, define new types of simple permutations, and analyze their interconnections and some asymptotic and geometrical properties of these classes.
Tomescu and Imran [Graphs and Combinatorics, 2010] proposed the following conjecture : for every ... more Tomescu and Imran [Graphs and Combinatorics, 2010] proposed the following conjecture : for every connected graph of order n the number of resolving pairs (i.e., pairs of vertices of G having distinct distances to all vertices of G) is bounded above by ⌊n 2 /4⌋ and solved into affirmative this assertion for graphs with diameter two. In this paper the conjecture is verified for bipartite graphs, graphs of order n and diameter n − 2 and for a subclass of graphs of diameter three. It is also shown that for every integers n,k such that n ≥ 3 and 2 ≤ k ≤ n − 1 there is a graph of order n and diameter k having ⌊n 2 /4⌋ resolving pairs.
Topological indices are numerical numbers assigned to the graph/structure and are useful to predi... more Topological indices are numerical numbers assigned to the graph/structure and are useful to predict certain physical/chemical properties. In this paper, we give explicit expressions of novel Banhatti indices, namely, first K Banhatti index B 1 G , second K Banhatti index B 2 G , first K hyper-Banhatti index HB 1 G , second K hyper-Banhatti index HB 2 G , and K Banhatti harmonic index H b G for hyaluronic acid curcumin and hydroxychloroquine. The multiplicative version of these indices is also computed for these structures.
Let G = (V, E) be a finite, simple and undirected graph having v = |V (G)| and e = |E(G)|. A grap... more Let G = (V, E) be a finite, simple and undirected graph having v = |V (G)| and e = |E(G)|. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2, . . . , 2q−1} such that, when each edge xy is assigned the label |f(x)−f(y)|, the resulting edge labels are {1, 3, 5, . . . , 2q−1}. Motivated by the work of Z. Gao [6], we have defined odd graceful labeling for some other union of graphs. In this paper we formulate odd-graceful labeling for disjoint unions of graphs consisting of generalized combs, ladder, star, bistar, caterpillar and path.
Kragujevac tree is indicated by K ; K ∈ K g q = s 2 t + 1 + 1 , s with order and size s 2 t + 1 +... more Kragujevac tree is indicated by K ; K ∈ K g q = s 2 t + 1 + 1 , s with order and size s 2 t + 1 + 1 and s 2 t + 1 , respectively. In this paper, we have a look at certain topological features of the total graph and line graph of the total graph of the considered tree, i.e ., Kragujevac tree, by computing different topological indices and polynomials.
An edge-magic total labeling of an (n,m)-graph G = (V,E) is a one to one map λ from V(G) ∪ E(G) o... more An edge-magic total labeling of an (n,m)-graph G = (V,E) is a one to one map λ from V(G) ∪ E(G) onto the integers {1,2,…,n + m} with the property that there exists an integer constant c such that λ(x) + λ(y) + λ(xy) = c for any xy ∈ E(G). It is called super edge-magic total labeling if λ (V(G)) = {1,2,…,n}. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edge-magic total labeling, called super edge-magic deficiency of a graph G, is denoted by μs(G) [4]. If such vertices do not exist, then deficiency of G will be + ∞. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].
A super edge-magic total (SEMT) labeling of a graph ℘(V, E) is a one-one map ϒ from V(℘)∪E(℘) ont... more A super edge-magic total (SEMT) labeling of a graph ℘(V, E) is a one-one map ϒ from V(℘)∪E(℘) onto {1, 2,…,|V (℘)∪E(℘) |} such that ∃ a constant “a” satisfying ϒ(υ) + ϒ(υν) + ϒ(ν) = a, for each edge υν ∈E(℘), moreover all vertices must receive the smallest labels. The super edge-magic total (SEMT) strength, sm(℘), of a graph ℘ is the minimum of all magic constants a(ϒ), where the minimum runs over all the SEMT labelings of ℘. This minimum is defined only if the graph has at least one such SEMT labeling. Furthermore, the super edge-magic total (SEMT) deficiency for a graph ℘, signified as $\mu_{s}(\wp)$ is the least non-negative integer n so that ℘∪nK1 has a SEMT labeling or +∞ if such n does not exist. In this paper, we will formulate the results on SEMT labeling and deficiency of fork, H -tree and disjoint union of fork with star, bistar and path. Moreover, we will evaluate the SEMT strength for trees.
Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd... more Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2,. .. , 2q− 1} such that, when each edge xy is assigned the label |f (x)− f (y)| , the resulting edge labels are {1, 3, 5,. .. , 2q− 1} and f is called an odd graceful labeling of G. Motivated by the work of Z. Gao [6] in which he studied the odd graceful labeling of union of any number of paths and union of any number of stars, we have determined odd graceful labeling for some other union of graphs. In this paper we formulate odd-graceful labeling for disjoint unions of graphs consisting of generalized combs, stars, bistars and paths.
Let G be a connected graph and d(x, y) be the distance between the vertices x and y. A subset of ... more Let G be a connected graph and d(x, y) be the distance between the vertices x and y. A subset of vertices W = {w1,w2, · · · ,wk} is called a resolving set for G if for every two distinct vertices x, y ∈ V (G), there is a vertex wi ∈ W such that d(x,wi) 6= d(y,wi). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension dim(G). Let F be a family of connected graphs Gn : F = (Gn)n�1 depending on n as follows: the order |V (G)| = '(n) and limn!1'(n) = ∞. If there exists a constant C > 0 such that dim(G) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension; otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), then F is called a family with constant metric dimension. The metric dimension of some classes of plane graphs has been determined in [3], [4], [5], [10], [12], [15] and [22], while...
Let G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of v... more Let G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of vertices W={w 1 ,w 2 ,⋯,w k } is called a resolving set for G if for every two distinct vertices x,y∈V(G), there is a vertex w i ∈W such that d(x,w i )≠d(y,w i ). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension dim(G). Let ℱ be a family of connected graphs G n :ℱ=(G n ) n≥1 depending on n as follows: the order |V(G)|=φ(n) and lim n→∞ φ(n)=∞. If there exists a constant C>0 such that dim(G)≤C for every n≥1 then we shall say that ℱ has bounded metric dimension; otherwise ℱ has unbounded metric dimension. If all graphs in ℱ have the same metric dimension (which does not depend on n), then ℱ is called a family with constant metric dimension. The metric dimension of some classes of plane graphs has been determined in several papers, while metric dimension of some classes o convex polyt...
We extend results about critically k-colorable graphs to choosability and paintability (list colo... more We extend results about critically k-colorable graphs to choosability and paintability (list colorability and on-line list colorability). Using a strong version of Brooks' Theorem, we generalize Gallai's Theorem about the structure of the low-degree subgraph of critically k-colorable graphs, and introduce a more adequate lowest-degree subgraph. We prove lower bounds for the edge density of critical graphs, and generalize Heawood's Map-Coloring Theorem about graphs on higher surfaces to paintability. We also show that on a fixed given surface, there are only finitely many critically k-paintable/k-choosable/ k-colorable graphs, if k ≥ 6. In this situation, we can determine in polynomial time k-paintability, k-choosability and k-colorability, by giving a polynomial time coloring strategy for ''Mrs. Correct''. Our generalizations of k-choosability theorems also concern the treatment of non-constant list sizes (non-constant k). Finally, we use a Ramsey-type lemma to deduce all 2-paintable, 2-choosable, critically 3-paintable and critically 3-choosable graphs, with respect to vertex deletion and to edge deletion.
Abstract: Different ways to describe a permutation, as a sequence of integers, or a product of Co... more Abstract: Different ways to describe a permutation, as a sequence of integers, or a product of Coxeter generators, or a tree, give different choices to define a simple permutation. We recollect few of them, define new types of simple permutations, and analyze their ...
Abstract: The paper contains enumerative combinatorics for positive braids, square free braids, a... more Abstract: The paper contains enumerative combinatorics for positive braids, square free braids, and simple braids, emphasizing connections with classical Fibonacci sequence. The simple subgraph of the Cayley graph of the braid group is analyzed in the final part.
Nanofluids are a very productive etymology of intensifying the process of heat and mass transport... more Nanofluids are a very productive etymology of intensifying the process of heat and mass transport systems linked with the industrial and thermal engineering systems. Nanomaterials have effective thermal properties and various applications in our daily life like in heat transfer, electronic cooling systems, energy production and biomedicine and also in the food industry. Keeping the entire motivating potential ramifications of nanoparticles in mind, this work is visualized in the mathematical model developed to show the heat and mass transport behavior of swimming motile organisms in the existence of the magnetic field, heat conduction source, thermal radiation, chemical processes and viscous dissipation. The flow of mass and heat transport under consideration is governed by nonlinear partial differential equations (PDEs) transformed into ordinary differential equations (ODEs) by implementing an eminent method called similarity transform and then numerical results obtained through MA...
Kragujevac tree is indicated by K; K ∈ Kg q�s(2t+1)+1,s with order and size s(2t + 1) + 1 and s(2... more Kragujevac tree is indicated by K; K ∈ Kg q�s(2t+1)+1,s with order and size s(2t + 1) + 1 and s(2t + 1), respectively. In this paper, we have a look at certain topological features of the total graph and line graph of the total graph of the considered tree, i.e., Kragujevac tree, by computing different topological indices and polynomials.
UPB Scientific Bulletin, Series A: Applied Mathematics and Physics , 2017
Tomescu and Imran [Graphs and Combinatorics, 2010] proposed the following conjecture : for every ... more Tomescu and Imran [Graphs and Combinatorics, 2010] proposed the following conjecture : for every connected graph of order n the number of resolving pairs (i.e., pairs of vertices of G having distinct distances to all vertices of G) is bounded above by ⌊n 2 /4⌋ and solved into affirmative this assertion for graphs with diameter two. In this paper the conjecture is verified for bipartite graphs, graphs of order n and diameter n − 2 and for a subclass of graphs of diameter three. It is also shown that for every integers n,k such that n ≥ 3 and 2 ≤ k ≤ n − 1 there is a graph of order n and diameter k having ⌊n 2 /4⌋ resolving pairs.
It is called super edge-magic total labeling if .V .G// D f1; 2; : : : ; ng. Furthermore, if G ha... more It is called super edge-magic total labeling if .V .G// D f1; 2; : : : ; ng. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edgemagic total labeling, called super edge-magic deficiency of a graph G, is denoted by s .G/ [4]. If such vertices do not exist, then deficiency of G will be C1. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].
Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd... more Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2,. .. , 2q − 1} such that, when each edge xy is assigned the label |f (x) − f (y)|, the resulting edge labels are {1, 3, 5,. .. , 2q − 1}. Motivated by the work of Z. Gao [6], we have defined odd graceful labeling for some other union of graphs. In this paper we formulate odd-graceful labeling for disjoint unions of graphs consisting of generalized combs, ladders, stars, bistars, caterpillars and paths .
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