On BC-Subtrees in Multi-Fan and Multi-Wheel Graphs

: The BC-subtree (a subtree in which any two leaves are at even distance apart) number is the total number of non-empty BC-subtrees of a graph, and is deﬁned as a counting-based topological index that incorporates the leaf distance constraint. In this paper, we provide recursive formulas for computing the BC-subtree generating functions of multi-fan and multi-wheel graphs. As an application, we obtain the BC-subtree numbers of multi-fan graphs, r multi-fan graphs, multi-wheel (wheel) graphs, and discuss the change of the BC-subtree numbers between different multi-fan or multi-wheel graphs. We also consider the behavior of the BC-subtree number in these structures through the study of extremal problems and BC-subtree density. Our study offers a new perspective on understanding new structural properties of cyclic graphs.


Introduction
A topological index is a numerical graph invariant that can quantitatively characterize the properties of the corresponding structure. Topological indices in general have applications in numerous areas such as network design, compounds synthesis, pharmacology, and biology. Consequently a large number (approximately four hundred) of topological indices have been introduced during previous decades.
Compared to the subtree number index [22][23][24][25][26], the BC-subtree number index is a counting-based topological index that incorporates leaf distance constraint. First of all, a BC-tree is a tree in which any two leaves are at even distance apart. This important structure was introduced by F. Harary et al. [27,28]. The BC-subtree number index or simply the BC-subtree index was first introduced in [13], as the number of all non-empty BC-subtrees (subtrees that are also BC-trees) of a graph.
We are particularly interested in the enumeration of BC-subtrees. Using a generating functional approach proposed by Yan and Yeh in [12] (for general subtrees), Yang et al. [13] presented algorithms of enumerating BC-subtrees of trees. Later Yang et al. [41] solved

Preliminaries
Before introducing our main results there are quite some preparations to do. Let G = (V(G), E(G); f , g) be a weighted graph with vertex set V(G), edge set E(G), vertex-weight function f = ( f o (u), f e (u)) ( f o (u) and f e (u) are the odd, even weight of u ∈ V(G), respectively) and edge-weight function g = g(e) for e ∈ E(G). First we list the necessary notations and terminologies.

Basic Notations
• d G (u, v): the distance between u ∈ V(G) and v ∈ V(G).
• the ω e v weight of T 1 , denoted by ω e v (T 1 ), is defined as: g(e).
The odd, even generating function of S(G; v) are respectively defined as Similarly, for a given BC-subtree T 2 of a weighted graph G, we define and the BC-subtree generating function of G is Let η(G; v, odd) (resp. η(G; v, even)) denote the number of v odd -subtrees (resp. v evensubtrees) in S(G; v, odd) (resp. S(G; v, even)). Then

Facts
With the above notations we introduce some previously established results that will be used in our arguments.
Let T = (V(T), E(T); f , g) be a weighted tree of order n > 1 with vertex weight function f (u) = ( f o (u), f e (u)) for u ∈ V(T) and edge weight function g = g(e) for e ∈ E(T), assumew = v i be a pendant vertex andẽ = (w,ũ) the pendant edge of T, let T = (V(T ), E(T ); f , g ) of order n − 1 be the weighted tree constructed from T through "contracting"w as follows: V(T ) = V(T)\w, E(T ) = E(T)\ẽ, and (1) for any v s ∈ V(T ), and g (e) = g(e) for any e ∈ E(T ).
Lemma 1 ([13]). Following the above notations, we have F( Lemma 2 ([13]). Let P n be a path on n vertices with vertex weight function f (v) = (0, y) for all v ∈ V(P n ) and edge weight function g(e) = z for all e ∈ E(P n ). Then, ). Let K 1,n be a star on n + 1 vertices with vertex weight function f (v) = (0, y) for all v ∈ V(K 1,n ) and edge weight function g(e) = z for all e ∈ E(K 1,n ). Then, A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices.

Lemma 4 ([41]
). Let U n = (V(U n ), E(U n ); f , g) be a weighted unicyclic graph of order n with no pendant vertices, whose vertex weight function f (v) = (0, y) for all v ∈ V(U n ) and edge weight function g(e) = z for all e ∈ E(U n ). Then,

Observations
We now move on to establish some new observations to facilitate our discussion of the main results later.
Let T = (V(T), E(T); f , g) be a weighted tree on n (≥ 2) vertices, with the vertex weight function f (u) = ( f o (u), f e (u)) for u ∈ V(T) and the edge weight function g = g(e) for e ∈ E(T). Let T v be a subtree of T and define Choose a pendant vertexw(∈ L(T)∧ / ∈ V(T v )) and letẽ = (w,ũ) denote the pendant edge; • Update the odd, even weight ofũ , and edge weight with rule as described in Lemma 1; • Remove the vertexw, edgeẽ and set T : T\{w,ẽ}; • Repeat the contracting process until the remaining tree is the weighed tree . From Lemma 1 we have the following two observations as immediate consequences. We skip the repetitive details. Theorem 1. Given T v a subtree of T = (V(T), E(T); f , g) containing vertex v, and let T * v = (V(T * v ), E(T * v ); f * , g * ) be the weighted subtree defined above, then, the odd and even generating functions of S(T; T v ), denoted by F(T; f , g; T v , odd) and F(T; f , g; T v , even), respectively, are ; f * , g * ) be the weighted subtree obtained from T defined above, then, the BC-subtree generating function of T containing the subtree T v is where We now establish the following for general graphs. ( (F(G i ; f , g; c, even) − f e (c)) + F BC (G 2 ; f , g; c).

(6)
If the vertex weight function f (v) = (0, y) for v ∈ V(G), then we have Proof. We will prove the claimed expression by considering different cases of the BCsubtrees of G containing c: where (i) T 1 is the set of BC-subtrees in S BC (G; c) such that all edges of each BC-subtree are only in G 1 ; (ii) T 2 is the set of BC-subtrees in S BC (G; c) such that all edges of each BC-subtree are only in G 2 ; (iii) T 3 is the set of BC-subtrees in S BC (G; c) such that edges of each BC-subtree are in both G 1 and G 2 .
It is easy to see that the BC-subtree generating function of T 1 and T 2 are and F BC (G 2 ; f , g; c).
Next, note that where T 1 ∪ T 2 are the trees obtained from T 1 and T 2 by identifying the vertex c; are the trees obtained from T 3 and T 4 by identifying their vertex c.
From these cases we can obtain the BC-subtree generating function of BC-subtree in T 3 as The theorem thus follows.
Through similar analysis we can also obtain the odd and even generating functions of S(G; c). We skip the technical details. Theorem 4. Let graph G = (V(G), E(G); f , g) be obtained from G 1 and G 2 with the unique common vertex c (see Figure 1), with the vertex weight function f (v) = ( f o (v), f e (v)) for v ∈ V(G) and the edge weight function g(e) = z for e ∈ E(G). Then F(G; f , g; c, odd) =F(G 1 ; f , g; c, odd) + F(G 2 ; f , g; c, odd) and F(G; f , g; c, even) If the vertex weight function f (v) = (0, y) for v ∈ V(G), then we have F(G; f , g; c, odd) =F(G 1 ; f , g; c, odd) + F(G 2 ; f , g; c, odd) and F(G; f , g; c, even) = 1 y

Further Definitions
We now introduce the graph structures under consideration in this paper.
Recall that K 1,n is a star on n + 1 vertices, P n is a path on n vertices, and C n is a cycle on n vertices. A fan graph, denoted by F n+1 , is a graph formed by adding an additional vertex adjacent to every vertex of P n . A wheel graph W n+1 is a graph formed from a cycle C n by adding a vertex adjacent to every vertex of C n .
Assume G 1 and G 2 are two disjoint graphs, a graph G = G 1 + G 2 is called the disjoint union of G 1 and . Moreover, let the product G 1 × G 2 denote the graph obtained from G 1 + G 2 by adding edges (a, b) with a ∈ V(G 1 ) and b ∈ V(G 2 ). In the specifal case when G 2 is a single vertex c, we write G 1 × G 2 as G 1 × c.
. . , k) be k distinct paths on l i vertices, and each P l i has n i copies with k ∑ i=1 n i l i = n, then, the graph (n 1 P l 1 + n 2 P l 2 + · · · + n k P l k ) × c 0 is called a multi-fan graph, where n 1 P l 1 + n 2 P l 2 + · · · + n k P l k is the disjoint union of k ∑ i=1 n i paths (n i is the number of paths of length l i − 1) and c 0 is the center vertex (see Figure 2).
Clearly, in the case of k = 1 and n 1 = 1, the multi-fan graph is just the fan graph F n+1 . For convenience we call the subgraph P l i × c 0 (i = 1, 2, . . . , k) of multi-fan graph (n 1 P l 1 + n 2 P l 2 + · · · + n k P l k ) × c 0 the sub-fan graph F l i +1 . It is easy to see that the fan graph F 1 is the single vertex c 0 for the case l i = 0, and F 2 is an edge for the case l i = 1. Figure 2. The multi-fan graph (iP 1 + P n j +1 + · · · + P n k ) × c 0 .
. . , k) be k distinct cycles on l i vertices. Suppose each C l i has n i copies with k ∑ i=1 n i l i = n. Then, the graph is called a multi-wheel graph, where n 1 C l 1 + n 2 C l 2 + · · · + n k C l k is the disjoint union of k ∑ i=1 n i cycles (n i is the number of cycles of length l i ) and c 0 is the center vertex. Clearly, in the case of k = 1 and n 1 = 1, the multi-wheel graph is just the wheel graph W n+1 . Similarly, we call the subgraph C l i × c 0 (i = 1, 2, . . . , k) of multi-wheel graph (n 1 C l 1 + n 2 C l 2 + · · · + n k C l k ) × c 0 the sub-wheel graph W l i +1 . Definition 3. Let G = (n 1 P l 1 + n 2 P l 2 + · · · + n k P l k ) × c 0 be the multi-fan graph with k ∑ i=1 n i l i = n. Then we also call G the r (1 ≤ r ≤ n, r is an integer) multi-fan graph, and is denoted by F r n+1 . If (n mod r) = 0, then let k = 2, l 1 = r, n 1 = n r , l 2 = 1, n 2 = (n mod r), and we call F r n+1 the r quasi-regular multi-fan graph.
Otherwise, let k = 1, l 1 = r, n 1 = n r , and we call F r n+1 the r regular multi-fan graph.

BC-Subtree Generating Functions of Multi-Fan Graphs
We now move on to study the BC-subtree generating functions. First we consider the multi-fan graphs in this section.
Theorem 5. Let G = (n 1 P l 1 + n 2 P l 2 + · · · + n k P l k ) × c 0 be a multi-fan graph with center vertex c 0 , the vertex weight function f (v) = (0, y) for v ∈ V(G) and the edge weight function g(e) = z for e ∈ E(G). For convenience we denote F(F l i +1 ; f , g; c 0 , odd) by F odd F l i +1 (c 0 ) and F(F l i +1 ; f , g; c 0 , even) by F even F l i +1 (c 0 ). Then and F even and Proof. First we consider two cases for the BC-subtrees of multi-fan graph G = (n 1 P l 1 + n 2 P l 2 + · · · + n k P l k ) × c 0 : (i) ones not containing the center c 0 ; (ii) ones containing the center c 0 .
With Lemma 2, we have the BC-subtree generating function of case (i) as With slightly more complicated structure analysis, Theorems 3 and 4, we have the BC-subtree generating function of case (ii) as Similarly, the odd and even generating functions, the BC-subtree generating functions of multi-fan graph (n i P l i ) × c 0 (i = 1, 2, . . . , k) containing c 0 are, respectively, We now label the non-center vertices of the sub-fan graph F l i +1 = P l i × c 0 (i = 1, 2, . . . , k) as c 1 , c 2 , . . . , c l i in counterclockwise order. In what follows, we further focus on computing the odd and even generating functions, and the BC-subtree generating functions of F l i +1 (i = 1, 2, . . . , k) containing c 0 .
Let e i = (c 0 , c l i ), then we have is the set of subtrees (resp. BC-subtrees) that contain c 0 , (c 0 , c l i ), but not (c l i −1 , c l i ); • S 3 (resp. S * 3 ) is the set of subtrees (resp. BC-subtrees) that contain c 0 , (c l i −1 , c l i ), but not (c 0 , c l i ); • S 4 (resp. S * 4 ) is the set of subtrees (resp. BC-subtrees) that contain c 0 , (c 0 , c l i ) and (c l i −1 , c l i ). We now study each case: is the tree obtained from T 1 , by attaching an edge e i at vertex c 0 ; (c) The set S 3 can be written as (d) Evidently, each T 4 ∈ S 4 must not contain the edge (c 0 , c l i −1 ). We further consider these subtrees by cases of containing edges (c 0 , for k = 1, 2, . . . , l i − 1, which can be rewritten as: By the definitions of ω o v weight, ω e v weight, odd, even generating function of subtrees containing a fixed vertex, (a)-(d), and Theorem 1, we have the followings.
Next we consider the BC-subtree generating function of some special cases of multifan graphs. First of all the BC-subtree generating function of the r multi-fan graph F r n+1 (1 ≤ r ≤ n, r is an integer) follows from Theorem 5. Theorem 6. Let F r n+1 (1 ≤ r ≤ n is a positive integer) be the r multi-fan graph defined in Definition 3 with vertex weight function f (v) = (0, y) for v ∈ V(F r n+1 ) and the edge weight function g(e) = z for e ∈ E(F r n+1 ), then and F BC (F 2 ; f , g; c 0 ) = 0, F odd F 1 (c 0 ) = 0, F even F 1 (c 0 ) = y.
Proof. Firstly, consider the BC-subtrees of multi-wheel graph G = (n 1 C l 1 + n 2 C l 2 + · · · + n k C l k ) × c 0 in two cases: (i) ones not containing the center c 0 ; (ii) ones containing the center c 0 .
From Lemma 4, we have the BC-subtree generating function for case (i) as Similar to the previous section, we have the BC-subtree generating function of case (ii) as Now label the l i non-center vertices of wheel graph W l i +1 = C l i × c 0 (i = 1, 2, . . . , k) with c 1 , c 2 , . . . , c l i . For convenience we let e t = (c 0 , c t ) (t = 1, 2, . . . , l i ), e * l i = (c 1 , c l i ) and e * l i −r = (c l i −r , c l i −r+1 ) (r = 1, 2, . . . , l i − 1). We partition the set of S(W l i +1 ; c 0 ) and S BC (W l i +1 ; c 0 ) into five cases: where • S 1 (resp. S * 1 ) is the set of subtrees (resp. BC-subtrees) that contain c 0 , but not (c 1 , c l i ); • S 2 (resp. S * 2 ) is the set of subtrees (resp. BC-subtrees) that contains c 0 and (c 1 , c l i ), but neither (c 0 , c l i ) nor (c l i −1 , c l i ); is the set of subtrees (resp. BC-subtrees) that contain c 0 , (c 0 , c l i ) and (c 1 , c l i ), but not (c l i −1 , c l i ); • S 4 (resp. S * 4 ) is the set of subtrees (resp. BC-subtrees) that contain c 0 , (c 0 , c l i ), (c l i −1 , c l i ) and (c 1 , c l i ); where • S l i −k,1 (resp. S * l i −k,1 ) is the set of subtrees (resp. BC-subtrees) that contain c 0 ∪ k r=0 e * l i −r , but not e l i −k or e * l i −k−1 ; • S l i −k,2 (resp. S * l i −k,2 ) is the set of subtrees (resp. BC-subtrees) that contain c 0 ∪ k r=0 e * l i −r and e l i −k , but not e * l i −k−1 ; • S l i −k,3 (resp. S * l i −k,3 ) is the set of subtrees (resp. BC-subtrees) that contain c 0 ∪ k r=0 e * l i −r , e l i −k and e * l i −k−1 ; • S l i −k,4 (resp. S * l i −k,4 ) is the set of subtrees (resp. BC-subtrees) that contain c 0 ∪ k r=0 e * l i −r and e * l i −k−1 , but not e l i −k . Again, we have that W (c 0 , c l i −r ).

The Behavior of the BC-Subtrees
With the theoretical foundation that was established in the previous sections, we will study the behavior of the BC-subtree number in the multi-fan and multi-wheel graphs. We first mention an extremal result as a simple consequence. We also briefly discuss the change of the BC-subtree numbers between different multi-fan or multi-wheel graphs. Lastly we consider the BC-subtree density in these structures.

BC-Subtree Number of F r n+1
The extremely problems with respect to a topological index concerns finding the extremal structures, among a given class of graphs, that maximize or minimize the index. These graphs always possess the best or worst of some desired properties [48][49][50]. We first point out the following simple fact. Proposition 1. The F 1 n+1 has 2 n − n − 1 BC-subtrees, fewer than any other F r n+1 (r = 1); the F n n+1 has more BC-subtrees than any other F r n+1 (r = n).
It is often interesting to know, when studying extremal problems, which graph structures have the second or third largest or smallest values of a certain index. To shed some light on this we ran some simulation, whose result is shown in Figure 3, with Cartesian and semi-log (Log-Y) coordinate, respectively.  From Figure 3a, it appears that among all F r n+1 (2 ≤ r ≤ n − 1), F n−1 n+1 seems to have the second largest BC-subtree number, and the F n−2 n+1 seems to be the third largest BC-subtree number for odd (or sufficient large) n.
It is also interesting to observe the change of the BC-subtree number as r changes, with the "local minimum" spread out for smaller values of r From Corollary 4 we can obtain data of similar nature for W n+1 (n ≥ 3). Figure 4 confirms the simple fact that the BC-subtree numbers of W n+1 increase very fast. The asymptotic expression of the BC-subtree number of W n+1 seems to be η BC (W n+1 ) ≈ exp(0.0909 + 0.893n). This is something that can be further verified through analytic combinatorics approaches.

BC-Subtree Density
The generating function approach provides us with much more information than just the BC-subtree number. In particular, we will examine the BC-subtree density here, for r multi-fan graph F r n+1 (1 ≤ r ≤ n, r is an integer) and W n+1 , respectively. For some of the work on this topic one may see [40]. First it is easy to see that n(F r n+1 ) = n(W n+1 ) = n + 1 Letting y = 1 in Theorem 6 and Theorem 8, we could obtain the so called edge generating function of BC-subtrees of F r n+1 and W n+1 , respectively, i.e., F BC (F r n+1 ; (0, 1), z) and F BC (W n+1 ; (0, 1), z)).
By the definition of BC-subtree density and Equation (83), the BC-subtree density of G * (G * = F r n+1 or W n+1 ) is simply We may now provide the BC-subtree densities of F r n+1 (1 ≤ r ≤ n) in Figure 5, with related data in Table 2.  Similarly, with Theorem 8 we obtain the generating function F BC (W n+1 ; (0, 1), z). Together with Equation (84) we can obtain the BC-subtree densities of W n+1 (see Figure 6 and Table 3).  From Table 2 and Figure 5, we see that F 1 22+1 , F 2 22+1 and F 3 22+1 ranks the first to the third with the smallest BC-subtree density, respectively. From Table 3 and Figure 6, we see that the BC-subtree density of W n+1 (3 ≤ n ≤ 24) maximized at n = 4, and decreases gradually when n ≥ 6.

Concluding Remarks
Motivated from the past studies of BC-trees, BC-subtrees, as well as the applications of structural properties of network graphs [51], we provide recursive formulae for computing the BC-subtree generating functions of multi-fan and multi-wheel graphs, and also derive the BC-subtree numbers of multi-fan graphs, r multi-fan graphs, multi-wheel (wheel) graphs. Moreover, the behavior of the BC-subtree numbers between different multi-fan or multi-wheel graphs, and extremal problems and BC-subtree density are also briefly discussed. These findings are likely useful in further understanding the properties and behaviors of these graphs. For future work it would be interesting to consider other well known topological indices on the multi-fan and multi-wheel graphs and compare their behaviors with that of the BC-subtree number.

Conflicts of Interest:
The authors declare no conflict of interest.