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A formula for the number of spanning trees of a multi-star related graph
Abstract
Using a new labeling technique and matrix computations, this paper derives a closed formula for the number of spanning trees of a multi-star related graph G = Kn − Km(a1,a2,…,am), where Km(a1,a2,…,am) consists of m star graphs such that the ith one has a root node connected to ai leaves, and further, the m roots are connected together to form a complete graph. This result generalizes the previous result by Nikolopoulos and Rondogiannis (1998) which is limited to m = 2, 3, 4.
... Theorem 2.13(Mao and Liu, [21]) The number r O (Γ) of non-isomorphic rooted maps on orientable surfaces underlying a simple graph Γ is ...
... An undirected circulant graph C n (a 1 , a 2 , a 3 , · · · , a k ) is a regular graph whose set of vertices is V = {0, 1, 2, [dots, n−1} and whose set of edges is E = {i, i+a i (mod n)/i = 0, 1, 2, · · · , n−1, j = 1, 2, · · · , k}. If a k ≤ n 2 , then C n (a 1 , a 2 , a 3 , · · · , a k ) is a 2k-regular graph; if a k = n 2 , then it is a 2k − 1-regular one, see Nikolopoulos [20] and Papadopoulos [21]. The well known formula τ (C n (1, 2)) = nF 2 n , where F n is the n th Fibonacci number, see Kleiman, and Golden [22]. ...
... Math.Combin.Book Ser. Vol.4(2019),[19][20][21][22][23][24][25][26][27][28] ...
The mathematical combinatorics is a subject that applying combinatorial notion to all mathematics and all sciences for understanding the reality of things in the universe, motivated by CC Conjecture of Dr.Linfan MAO on mathematical sciences. The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly, which publishes original research papers and survey articles in all aspects of mathematical combinatorics, Smarandache multi-spaces, Smarandache geometries, non-Euclidean geometry, topology and their applications to other sciences.
... Many cases have been examined depending on the choice of G. For example, there exist closed formulas for the cases where G is is a pairwise disjoint set of edges [24], a chain of edges [16], a cycle [11], a star [20], a multi-star [19,25], a multi-complete/star graph [7], a labelled molecular graph [5], and more recent results when G is a circulant graph [12,26], a quasithreshold graph [18], and so on (see Berge [2] for an exposition of the main results). ...
... , b r ), consists of a complete graph K r with vertices labelled v 1 , v 2 , . . . , v r , and b i vertices of degree one, which are incident with vertex v i , 1 ≤ i ≤ r [7,19,25]; see Figure 1(b). Let G = K r (b 1 , b 2 , . . . ...
... , b r ) be a multi-star graph on p = r + b 1 + b 2 + · · · + b r vertices. It has been proved [7,19,25] that the number of spanning trees of the graph K n − G is given by the following closed formula: ...
International audience
The K_n-complement of a graph G, denoted by K_n-G, is defined as the graph obtained from the complete graph K_n by removing a set of edges that span G; if G has n vertices, then K_n-G coincides with the complement øverlineG of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K_n^m #x00b1 G, where K_n^m is the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K_n^m; the graph K_n^m + G (resp. K_n^m - G) is obtained from K_n^m by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K_n^m by adding and removing edges of multigraphs spanned by sets of edges of the graph K_n^m. We also prove closed formulas for the number of spanning tree of graphs of the form K_n^m #x00b1 G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.
... For any subgraph H of the complete graph K n , the K n -complement of H, denoted by K n − H, is defined as the graph obtained from K n by removing the edges of H; note that if H has n vertices then K n − H coincides with the complement H of H. Many different types of graphs K n − H have been examined: for example, there exist closed formulas for the cases where H is a pairwise-disjoint set of edges [22], a chain of edges [13], a cycle [7], a star [19], a multi-star [18,23], a multi-complete/star graph [4], a labeled molecular graph [3], and more recently if H is a circulant graph [8,12,24], a quasi-threshold graph [17], and so on (see Berge [1] for an exposition of the main results). ...
... A common approach for determining the number of spanning trees of a graph G relies on a classic result known as the complement-spanning-tree matrix theorem [21], which expresses the number of spanning trees of G as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of G, i.e., adjacency matrix, adjacency lists, etc. Calculating the determinant of the complement-spanning-tree matrix seems to be a promising approach for computing the number of spanning trees of families of graphs of the form K n − H, where H is a graph that exhibits symmetry (see [1,4,7,18,17,16,23,24]). ...
... Multi-star graph [18,23] m ≥ 1, appropriate α = 0, β = γ = δ = 0 ...
In this paper we introduce the class of graphs whose complements are asteroidal (star-like) graphs and derive closed formulas for the number of spanning trees of its members. The proposed results extend previous results for the classes of the multi-star and multi-complete/star graphs. Additionally, we prove maximization theorems that enable us to characterize the graphs whose complements are asteroidal graphs and possess a maximum number of spanning trees.
... Many cases have been examined depending on the choice of G. For example, there exist closed formulas for the cases where G is is a pairwise disjoint set of edges [24], a chain of edges [16], a cycle [11], a star [20], a multi-star [19,25], a multi-complete/star graph [7], a labelled molecular graph [5], and more recent results when G is a circulant graph [12,26], a quasithreshold graph [18], and so on (see Berge [2] for an exposition of the main results). ...
... , b r ), consists of a complete graph K r with vertices labelled v 1 , v 2 , . . . , v r , and b i vertices of degree one, which are incident with vertex v i , 1 ≤ i ≤ r [7,19,25]; see Figure 1(b). Let G = K r (b 1 , b 2 , . . . ...
... , b r ) be a multi-star graph on p = r + b 1 + b 2 + · · · + b r vertices. It has been proved [7,19,25] that the number of spanning trees of the graph K n − G is given by the following closed formula: ...
We presentan efficientalgorithm for determining the number of spanning trees in the class of P
4-reducible graphs which are perfect graphs and generalize both the well-known class of cographs and the class of quasi-threshold graphs. In particular
for a P
4-reducible graph G on n vertices and m edges our algorithm computes the number of spanning trees of G in O(n+m) time and space, wherethe complexity of arithmetic operations is measured under the uniform cost criterion. The algorithm takes advantage of the modular decomposition tree of the input graph which it gradually shrinks in asystematic fashion until it becomes a single vertex while at the same time appropriately updating certain parameters whose product gives the desired number of spanning trees. The correctness of the algorithm is established through the Kirchhoff matrix tree theorem and is also based on structural and algorithmic properties of the graphs with few P
4s. Our results generalize previous results and extend the family of graphs admitting linear-time algorithms for the number of their spanning trees.
... Many cases have been examined depending on the choice of G. For example, there exist closed formulas for the cases where G is is a pairwise disjoint set of edges [24], a chain of edges [16], a cycle [11], a star [20], a multi-star [19,25], a multi-complete/star graph [7], a labelled molecular graph [5], and more recent results when G is a circulant graph [12,26], a quasithreshold graph [18], and so on (see Berge [2] for an exposition of the main results). ...
... , b r ), consists of a complete graph K r with vertices labelled v 1 , v 2 , . . . , v r , and b i vertices of degree one, which are incident with vertex v i , 1 ≤ i ≤ r [7,19,25]; see Figure 1(b). Let G = K r (b 1 , b 2 , . . . ...
... , b r ) be a multi-star graph on p = r + b 1 + b 2 + · · · + b r vertices. It has been proved [7,19,25] that the number of spanning trees of the graph K n − G is given by the following closed formula: ...
In this paper we examine the classes of graphs whose Kn-complements are trees and quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of Kn , the Kn-complement of H is the graph Kn H which is obtained from Kn by removing the edges of H . Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of the graph. Our results generalize previous results and extend the family of graphs of the form Kn H admitting formulas for the number of their spanning trees.
... Many cases have already been examined. For example there exist formulas for the cases when H is a pairwise disjoint set of edges [20], when it is a star [17], when it is a complete graph [1], when it is a path [5], when it is a cycle [5], when it is a multi-star [3,16,22], and so on (see Berge [1] for an exposition of the main results). ...
... A QT-graph is a graph that contains no induced subgraph isomorphic to P 4 or C 4 , the path or cycle on four vertices [7,12,15,21]. Our proofs are based on a classic result known as the complement spanning-tree matrix theorem [19], which expresses the number of spanning trees of a graph G as a function of the determinant of a matrix that can be easily constructed from the adjacency relation (adjacency matrix, adjacency lists, etc.) of the graph G. Calculating the determinant of the complement spanning-tree matrix seems to be a promising approach for computing the number of spanning trees of families of graphs of the form K n − H, where H posses an inherent symmetry (see [1,3,5,16,22,23]). In our cases, since neither trees nor quasi-threshold graphs possess any symmetry, we focus on their structural and algorithmic properties. ...
In this paper we examine the classes of graphs whose K-n-complements are trees or quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of K-n, the K-n-complement of H is the graph K-n-H which is obtained from K-n by removing the edges of H. Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of the graph. Our results generalize previous results and extend the family of graphs of the form K-n-H admitting formulas for the number of their spanning trees.
... There has been much interest in deriving formulas regarding the number of spanning trees of the K n -complements of different types of graph G. For instance, see the results for the cases when G is pK 2 [31], a star [26], a complete graph [2], a path [20], a cycle [15], a multi-star [6,23,34], a complete multipartite graph [27], a circular graph [35], and a quasi-threshold graph [21], respectively. In this paper, we focus on counting spanning trees of the K n -complements of a bipartite graph. ...
For a subgraph G of a complete graph Kn, the Kn-complement of G, denoted by Kn-G, is the graph obtained from Kn-G by removing all the edges of G. In this paper, we express the number of spanning trees of the Kn-complement Kn-G of a bipartite graph G in terms of the determinant of the biadjcency matrices of all induced balanced bipartite subgraphs of G, which are nonsingular, and we derive formulas of the number of spanning trees of Kn-G for various important classes of bipartite graphs G, some of which generalize some previous results.
... Many cases have been examined depending on the choice of G. It has been studied when G is labelled molecular graph [1], when G is a circulant graph [2], when G is a complete multipartite graph [3], when G is a cubic cycle and quadruple cycle graph [4], when G is a threshold graph [5] and so on. A spanning tree of G is a minimal connected subgraph of G that has the same vertex set as G. ...
In mathematics, one always tries to get new structures from given ones. This also applies to the realm of graphs, where one can generate many new graphs from a given set of graphs. In this paper, we derive simple formulas of spanning trees of some families of graphs generated by triangle using linear algebra and the knowledge of difference equations. Finally, we compare the entropy of our graphs with other studied graphs with the average degree being 4 and 6. ARTICLE HISTORY
... The problem of computing the number of spanning trees on the graph G is an important problem in graph theory. In this context, there are a lot of papers that derive formulas and algorithms (see [7,8,9,10]). In particular, if K m,n is the complete bipartite graph, it is very well known that the number of spanning tree is equal to m n−1 n m−1 [6]. ...
In this paper we give a linear time algorithm for computing the number of spanninig trees in double nested graphs.
... The number of spanning trees of some special network has been taken into evaluation [11][12][13][14][15][16][17][18][19][20]. Recently, some authors derived results about the counting where the number of spanning trees can be found from [21][22][23][24][25][26][27][28][29]. ...
The number of spanning trees in graphs or in networks is an important issue. The evaluation of this number not only is interesting from a mathematical (computational) perspective but also is an important measure of reliability of a network or designing electrical circuits. In this paper, a simple formula for the number of spanning trees of the Cartesian product of two regular graphs is investigated. Using this formula, the number of spanning trees of the four well-known regular networks can be simply taken into evaluation.
... Many cases have been examined depending on the choice of G. For example, there exist closed formulas for the cases where G is a pairwise disjoint set of edges [11], when it is a chain of edges [6], a cycle [5], a star [10], a multistar [8,12], a complete graph [1], a multi-complete/star graph [3], a quasi-threshold graph [9], and so on (see Berge [1] for an exposition of the main results). ...
In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {G n } be a sequence of graphs G 0 , G 1 , G 2 , . . . that belong to a particular class. We consider graphs of the form K n − G n that result from the complete graph K n after removing a set of edges that span G n . We study the spanning tree behaviour of the sequence {K n − G n } when n → ∞ and the number of edges of G n scales according to n. More specifically, we define the spanning tree indicator α({G n }), a quantity that characterizes the spanning tree behaviour of {K n − G n }. We derive closed formulas for the spanning tree indicators for certain well-known classes of graphs. Finally, we demonstrate that the indicator can be used to compare the spanning tree behaviour of different classes of graphs (even when their members never happen to have the same number of edges). 2004 Elsevier B.V. All rights reserved.
... Weinberg's results have also been generalized in [22]. Closed formulae also exist for the cases where S is a star [20], a complete k-partite graph [21], a multi-star [19,25], and so on. The number of spanning trees in the complement graph is investigated in [13,16] when the graph with maximal number of spanning trees is studied. ...
Kirchhoff 's Matrix Tree Theorem permits the calculation of the number of spanning trees in any given graph G through the evaluation of the determinant of an associated matrix. In the case of some special graphs Boesch and Prodinger [Graph Combin. 2 (1986) 191–200] have shown how to use properties of Chebyshev polynomials to evaluate the associated determinants and derive closed formulas for the number of spanning trees of graphs.In this paper, we extend this idea and describe how to use Chebyshev polynomials to evaluate the number of spanning trees in G when G belongs to one of three different classes of graphs: (i) when G is a circulant graph with fixed jumps (substantially simplifying earlier proofs), (ii) when G is a circulant graph with some non-fixed jumps and when (iii) G=Kn±C, where Kn is the complete graph on n vertices and C is a circulant graph.
The enumeration of spanning trees in a finite graph is an important problem related to various domains of mathematics, physics and network reliability that has been investigated by many researchers. A network N is called a closed chain of planar networks if its modelling is defined by n planar graphs connected by articulation points. Some recent studies focus on finding the number of spanning trees in the star flower network which is a closed chain of cycles. In this paper, we give a general method counting the number of spanning trees in a closed chain of planar networks. We propose a recursive approach based on the contraction of nodes and as application, we obtain explicit expressions for the number of spanning trees in the closed chain of maximal planar networks and the pseudofractal scale-free network.
We present a new simple linear-time algorithm for determining the number of spanning trees in the class of complement reducible graphs, also known as cographs. For a cograph G on n vertices and m edges, our algorithm computes the number of spanning trees of G in O(n+m) time and space, where the complexity of arithmetic operations is measured under the uniform cost criterion. The algorithm takes advantage of the cotree of the input cograph G and works by contracting it in a bottom-up fashion until it becomes a single node. Then, the number of spanning trees of G is computed as the product of a collection of values which are associated with the vertices of G and are updated during the contraction process. The correctness of our algorithm is established through the Kirchhoff matrix tree theorem, and also relies on structural and algorithmic properties of the class of cographs. We also extend our results to a proper superclass of cographs, namely, the P 4 -reducible graphs, and show that the problem of finding the number of spanning trees of a P 4 -reducible graph has linear-time solution.
the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K, n卤 G, where K, nis the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K, n; the graph K, n+ G (resp. K, n-G) is obtained from K, nby adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K, nby adding and removing edges of multigraphs spanned by sets of edges of the graph K, n. We also prove closed formulas for the number of spanning tree of graphs of the form K, n卤 G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.
In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node; then, the number of spanning trees of G is computed as the product of a collection of values which are associated with the vertices of G and are updated during the contraction process. In particular, when applied on a (q,q − 4)-graph for fixed q, a P
4-tidy graph, or a tree-cograph, our algorithm computes the number of its spanning trees in time linear in the size of the
graph, where the complexity of arithmetic operations is measured under the uniform-cost criterion. Therefore we give the first
linear-time algorithm for the counting problem in the considered graph classes.
This paper derives a closed formula for the number of spanning trees of a multi-complete/star related graph G=Kn−Km(a1,a2,…,al;b1,b2,…,bm−l), where Km(a1,a2,…,al;b1,b2,…,bm−l) consists of l complete graphs and m−l star graphs such that the ith complete graph has ai+1 nodes; the jth star graph has bj+1 nodes, and further, the related m roots are connected together to form a complete graph. The proposed results extend previous results to a larger graph class. In addition, we provide a general maximization theorem for the multi-star graph.
In this paper we propose a linear-time algorithm for determining the number of spanning trees in cographs; we derive formula for the number of spanning trees of a cograph G on n vertices and m edges, and prove that the problem of counting the number of spanning trees of G can be solved in O(n+m) time. Our proofs are based on the Kirchho matrix tree theorem which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily construct from the adjacency relation of the graph. Our results generalize previous results regarding the number of spanning trees.
In this paper we compute the number of spanning trees of a specific family of graphs using techniques from linear algebra and matrix theory. More specifically, we consider the graphs that result from a complete graph Kn after removing a set of edges that spans a multi-star graph Km(a1, a2,…, am). We derive closed formulas for the number of spanning trees in the cases of double-star (m = 2), triple-star (m = 3), and quadruple-star (m = 4). Moreover for each case we prove that the graphs with the maximum number of spanning trees are exactly those that result when all the ais are equal.
Theorems are proved for each of two models representing purely repulsive interactions, the `Gaussian' and the `rigid line'. In both cases we study the fugacity series by starting with the complete diagram of l points connected by ½l(l-1) lines. For the Gaussian model, it is proved rigorously that the cluster integral corresponding to any diagram can be expressed as a product, each disjoint set of lines in the complementary diagram contributing one factor. The result for the `rigid-line' model is that any cluster integral can be expressed as a sum, each disjoint set of lines in the complementary diagram contributing a term, with corresponding results for the `rigid-square' and `rigid-cube' models. These results enable the consequences of the `Gaussian' model to be worked out almost completely, and provide some justification for approximate treatments that neglect all but the more `open' diagrams. The `rigid-square' model is more difficult analytically, and only a few preliminary deductions have been made.
In this paper we exhibit an explicit formula for the number of trees in a certain type of network. In particular, we define, in Definition 3, the concept of an incompletely partitioned network, and offer in Theorem 9 a formula for the number of trees of an incompletely partitioned network. Our formulas include, as special cases, the formulas exhibited previously in the literature by Weinberg [1] and Bedrosian [2].
The number of trees in a certain network, Notices Amer
- Neil
P.V. O’Neil, The number of trees in a certain network, Notices Amer. Math. Sot. 10 (1963) 569.