On Enumeration of some Non-Isomorphic Dendroids

A dendroid is a connected semigraph without a strong cycle. In this paper, we obtain the various results on the enumeration of the non-isomorphic dendroids containing two edges and the dendroids with three edges.


Introduction
Graph theory [1] is a study of some 2-tuple (u, v) of distinct elements belonging to a set of vertices V , with a condition that for each such 2-tuple, (u, v) = (v, u). Semigraph theory [2] is a generalization of graph theory and is a study of n-tuples of distinct elements belonging to a set V of vertices. Recently, many interesting generalizations of the concepts of graph theory have been obtained in semigraph. [3,4,5,6].
It is a well-known fact that the enumeration of graphs is one of the most important areas of graph theory and is applied in chemistry, physics, biology, information theory and so on [7,8,9]. Semigraphs applications are nicely explored in [10,11,12]. Due to the number of varieties at each and every step of semigraph concepts, the problem of generalizing the graph-enumeration for all the semigraphs appears to be np-complete. Therefore, some graph theorist have obtained the enumeration of some special categories of semigraphs. The enumeration of some edge complete semigraphs have been studied by K. Kayathri and S. Pethanachi Selvam [13,14] and the enumeration of labeled semigraphs containing non-adjacent s-edges is obtained by authors of the present paper [15].
Arthur Cayley is one of the pioneers of graph theory, known for his work on counting trees. His Cayley's Formula [7] gives the number of labeled trees on n-vertices. The concept of tree is generalized as a dendroid in a semigraphs. The structure of many chemical compounds of more than two elements can be represented as dendroids which otherwise not possible in case of trees. The aim of present work is to obtain the enumerations of such non-isomorphic dendroids containing two edges and the dendroids containing three edges. We expect to see many applications of this in the future. A dendroid is a connected semigraph without strong cycles. In Figure 2 , semigraph T is a dendroid.

Main Results
Here, we establish various results on the enumeration of dendroids with two edges and the dendroids with three edges. The Theorem 12 is also proved in [15], here we give an alternate proof to develop the proofs of remaining Theorems. Theorem 12. The number of non-isomorphic dendroids on n-vertices with exactly two edges of cardinalities k 1 2 and k 2 2 with 3 k 1 + k 2 − 1 n is Proof. Let G be a dendroid on n-vertices containing two edges E 1 = (u 1 , u 2 , . . . , u k 1 −1 , u k 1 ) and Let v be the common vertex between edges E 1 and E 2 and (i, j) represents the location of vertex v as i th position in E 1 and j th position in the edge E 2 . As an edge in a semigraph is symmetric, that is (u 1 , u 2 , . . . , u k−1 , u k ) = (u k , u k−1 , . . . , u 2 , u 1 ), the dendroids corresponding to the position (i, j) for i = 1, 2, 3, . . . , k 1 , j = 1, 2, 3, . . . , k 2 of v is isomorphic to the dendroids corresponding the position (k 1 − i + 1, k 2 − j + 1) for i = 1, 2, 3, . . . , k 1 , j = 1, 2, 3, . . . , k 2 of v. Henceforth, for this theorem as well as for the Theorem 13 and Theorem 14, we consider i = 1, 2, 3, . . . , ⌉ . Now, we consider the following two cases based on the cardinalities k 1 and k 2 of the edges E 1 and E 2 .
As k 1 ̸ = k 2 , the non-isomorphic dendroids of two edges E 1 and E 2 are obtained by selecting the i th position in E 1 by ⌈ k 1 2 ⌉ ways and the j th position in the edge E 2 by ⌈ k 2 2 ⌉ ways.
Hence, the number of non-isomorphic dendroids is

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Here, we consider the following different cases based on (i, j): ways and for each i, the j th position in the edge E 2 edge can be selected in ways and hence, the number of dendroids is . Hence, the number of non-isomorphic dendroids is . As k 1 = k 2 , for non-isomorphic dendroids, the (i, j) th position in E 1 and E 2 can be selected in ⌈ k 2 ⌉ ways. Hence, the number of non-isomorphic dendroids is Combining both the results, the number of non-isomorphic dendroids is

Theorem 13. The number of non-isomorphic dendroids on n-vertices with exactly three edges
if both k 1 and k 2 are even, , if k 1 is odd and k 2 is even, Proof. Let G be a dendroid on n-vertices containing three edges E 1 = (u 1 , u 2 , . . . , u k 1 −1 , u k 1 ) and Let v be the common vertex between the edges E 1 ,E 2 and (i, j) represents the location of a vertex v as i th position in E 1 and j th position in the edge E 2 . As dendroid with exactly three edges, can be obtained by adding a third edge to the dendroid of two edges, the number of dendroids on three edges will depend on the number of dendroids on two edges and also the position (i, j) of v in both edges. Therefore, we consider the following different cases: Case (i) Let both k 1 and k 2 are even.
In this case, from Theorem 12, the number of dendroids on two edges E 1 , E 2 of cardinality k 1 and k 2 for k 1 For each of these dendroids, as k 1 and k 2 (k 1 ̸ = k 2 ) are even, the third edge can be added by choosing one vertex from the (k 1 + k 2 − 1) vertices of E 1 , E 2 and one vertex from Hence, the number of non-isomorphic dendroids is

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Case (ii) Without loss of generality, let k 1 be odd and k 2 be even.
As k 1 is odd, the number of dendroids depend on position of the common vertex v in the edge E 1 . Therefore, here the following two sub-cases will arise: . Clearly, the i th position in E 1 can be selected in ways and for each i, the j th position in the edge E 2 can be selected in . For each of these dendroids, the third edge can be added by choosing one of the vertices from the (k 1 + k 2 − 1) vertices of E 1 , E 2 and one vertex from Hence, the number of non-isomorphic dendroids with v at the non-centre position is Clearly, the i th position in E 1 can be selected in one way and the j th position in the edge E 2 edge can be selected in If v is at the centre position of the edge E 1 , then the dendroids obtained by adding a third edge by choosing the common vertices as u 1 , u 2 , . . . u (⌈ k 1 2 ⌉) of the edge E 1 is isomorphic to the dendroids obtained by adding a third edge by choosing the common vertex as u (⌈ k 1 2 ⌉) , u (⌈ k 1 2 ⌉+1) , . . . u k 1 of the edge E 1 . Hence, the non-isomorphic dendroids are obtained by adding a third edge by choosing a common vertex from either vertices of E 1 , E 2 and a vertex from ⌈ k 3 2 ⌉ to vertices of the edge E 3 . Therefore, the number of such non-isomorphic dendroids with exactly three edges is Combining both the results of (a) and (b), the number of non-isomorphic dendroids with k 1 odd and k 2 is even ( Case (iii) Let both k 1 and k 2 be odd. (k 1 ̸ = k 2 ).
As both k 1 and k 2 are odd, the number of dendroids depends on the (i, j) position of common vertex v of E 1 , E 2 in both the edges E 1 and E 2 . Therefore, here the following four sub-cases will arise: that is v be at the non-centre position of both the edges E 1 , E 2 . Let the common vertex v be located at (i, j), for i = 1, 2, . . . , in the edges E 1 , E 2 . Clearly, the i th position in E 1 can be selected in ways and for each i, the j th position in the edge E 2 edge can be selected in ways, hence the number of dendroids is

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For each of these dendroids, the third edge can be added by choosing one of the vertices from the (k 1 + k 2 − 1) vertices of E 1 , E 2 and one vertex from ⌈ k 3 2 ⌉ vertices of the edge E 3 . Hence, the number of non-isomorphic dendroids with v at the non-centre position is that is v be at the centre position of E 1 and the non-centre positions of E 2 . Let the common vertex v be located at (i, j), in the edges E 1 , E 2 . Clearly, i th position in E 1 can be selected in one way and the j th position in the edge E 2 edge can be selected in ways, hence the number of dendroids is If v is at the centre position of edge E 1 then the dendroids obtained by adding third edge by choosing common vertices as u 1 , u 2 , . . . u (⌈ k 1 2 ⌉) of edge E 1 , is isomorphic to the dendroids obtained by adding third edge by choosing common vertex as u (⌈ k 1 2 ⌉) , u (⌈ k 1 2 ⌉+1) , . . . u k 1 of edge E 1 . Hence, the non-isomorphic dendroids are obtained by adding third edge by choosing one common vertex from either first vertices of the edge E 3 . Therefore, the number of non-isomorphic dendroids with v at the centre position of E 1 and the non-centre position of that is the common vertex v be at the non-centre position of E 1 and the centre positions of E 2 . Interchanging the role of E 1 and E 2 in the above sub-case (b), we will get the number of non-isomorphic dendroids with v at centre position of E 2 and at non-centre position of that is v be at the centre position of E 1 and E 2 .
Let the common vertex v be located at (i, j), . In this case, the dendroids obtained by adding a third edge by choosing the common vertices as u 1 , u 2 , u 3 , is isomorphic to the dendroids obtained by adding a third edge by choosing the common vertex as . Hence, the non-isomorphic dendroids are obtained by adding a third edge by choosing one common vertex from either first vertices of E 1 , E 2 and one vertex from the ⌈ k 3 2 ⌉ vertices of the edge E 3 . Therefore, the number of non-isomorphic dendroids with v at the centre position of E 1 and

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Combining all the cases, the number of non-isomorphic dendroids with both k 1 and k 2 ( Theorem 14. The number of non-isomorphic dendroid on n-vertices with exactly three edges E1, E2, E3 of cardinality k 1 2, k 2 2 and k 3 2 ( Proof. Let G be a dendroid on n-vertices containing three edges E 1 = (u 1 , u 2 , . . . , u k 1 −1 , u k 1 ) and As dendroid with exactly three edges, can be obtained by adding a third edge to the dendroid of two edges, therefore the number of dendroid on three edges will depend on the number of dendroid on two edges and also the position of the common vertex in both edges. Let (i, j) represents the location of the common vertex v as i th in E 1 and j th in the edge E 2 . Therefore, we consider the following different cases based on (i, j) and cardinality k 1 = k 2 = k: Case (i) Let both k 1 and k 2 (k 1 = k 2 = k) are odd.
Here, we consider the following different cases based (i, j).
Let the common vertex v is located at (i, j), for i = j = 1, 2, . . .
. In this case , the dendroids obtained by adding a third edge by choosing the common vertex as one of the vertices amongst u 1 , u 2 , . . . , u k−1 , u k of edge E 1 is isomorphic to the dendroids obtained by adding a third edge by choosing the common vertex as one of the vertices amongst v 1 , v 2 , . . . , v k−1 , v k of edge E 2 . Hence, the non-isomorphic dendroids are obtained by adding a third edge by choosing one common vertex from either k vertices of edge E 1 or k vertices of E 2 and one vertex from the ⌈ k 3 2 ⌉ vertices of the edge E 3 . Therefore, the number of non-isomorphic dendroids is = k that is located at the centre of both the edges E 1 and E 2 . Let the common vertex v be located at . Then the number of such dendroids on two edges E 1 , E 2 of cardinality k 1 and k 2 for k 1 = k 2 = k is exactly ONE. In this case , the dendroids obtained by adding a third edge by choosing the common vertex as u i , i = 1, 2, . . .
of edge E 1 is isomorphic to the dendroids obtained by adding a third edge by choosing the common vertex as u k−i , i = 1, 2, . . .
, i ̸ = j is isomorphic to the dendroids corresponding to the common vertex v located at (j, i) for i, j = 1, 2, . . .
Hence, the number of non-isomorphic dendroids with common vertex v located at (i, j) as discussed above is 1 . Now the dendroids containing three edges can be obtained by adding a third edge to the dendroids of two edges, by choosing any one of the (2k − 1) vertices of 'E 1 or E 2 ' in (2k − 1) ways. Therefore, in this case, the number of non-isomorphic dendroids is that is located at the centre of the edges E 1 but not the centre of the edge E 2 . Let the common vertex v of E 1 , E 2 be located at . Hence, the number of non-isomorphic dendroids corresponding to v located at . In this case, the dendroids obtained by adding a third edge by choosing the common vertices as u i , i = 1, 2, . . . ⌈ k 2 ⌉ of edge E 1 is isomorphic to the dendroids obtained by adding a third edge by choosing the common vertex as u k−i , i = 1, 2, . . . ⌈ k 2 ⌉ of edge E 2 . Hence, the non-isomorphic dendroids are obtained by adding a third edge by choosing one common vertex from ⌈ k 2 ⌉ vertices of edge E 1 or k vertices of E 2 that is vertices of E 1 ,E 2 and one vertex from the ⌈ k 3 2 ⌉ vertices of the edge E 3 . Therefore, the number of non-isomorphic dendroids is that is located at centre of the edges E 2 but not the centre of the edge E 1 . As k 1 = k 2 = k, the dendroid corresponding to the common vertex located at (⌈ k 2 ⌉ , j ) for j = 1, 2, . . . Combining all the five results (a), (b), (c), (d), (e) of case (i), the number of non-isomorphic dendroids with k 1 = k 2 = k as odd number is