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Steiner Wiener Index of Block Graphs Steiner Wiener index of block graphs MatjaˇzKovˇse, Rasila V A, Ambat Vijayakumar Abstract Let S be a set of vertices of a connected graph G. The Steiner distance of S is the minimum size of a connected subgraph of G containing all the vertices of S. The Steiner k-Wiener index is the sum of all Steiner distances on sets of k vertices of G. Different simple methods for calculating the Steiner k-Wiener index of block graphs are presented. MR Subject Classifications: 05C12 Keywords: Distance in graphs; Steiner distance; Wiener index; Steiner k-Wiener index; block graphs; poset of block graphs. 1 Introduction All graphs in this paper are simple, finite and undirected. Unless stated otherwise let n = jV (G)j and m = jE(G)j, hence n denotes the order and m size of a graph G. If G is a connected graph and u; v 2 V (G), then the (geodetic) distance dG(u; v) (or simply d(u; v) if there is no confusion about G) between u and v is the number of edges on a shortest path connecting u and v. The Wiener index W (G) of a connected graph G is defined as X W (G) = d(u; v): fu;vg2V (G) The first investigation of this distance-based graph invariant was done by Wiener in 1947, who realized in [21] that there exist correlations between the boiling points of paraffins and their molecular structure and noted that in the case of a tree it can be easily calculated from the edge contributions by the following formula: X W (T ) = n(T1)n(T2); (1) e2E(T ) where n(T1) and n(T2) denote the number of vertices in connected components T1 and T2 formed by removing an edge e from the tree T . The Steiner distance of a graph has been introduced in [7] by Chartrand et al., as a natural generalization of the geodetic graph distance. For a connected graph G and S ⊆ V (G), the Steiner distance dG(S) (or simply d(S)) among the vertices of S is the minimum size among all arXiv:1805.08143v2 [math.CO] 13 Sep 2018 connected subgraphs whose vertex sets contain S. Note that any such subgraph H is a tree, called a Steiner tree connecting vertices from S. Vertices of S are called terminal vertices of tree H, while the rest of the vertices of H are called inner vertices of the Steiner tree H. If S = fu; vg, then d(S) = d(u; v) coincides with the geodetic distance between u and v. In [9] Dankelmann et al. followed by studying the average k-Steiner distance µk(G). In [15], Li et al. introduced a generalization of the Wiener index by using the Steiner distance. The Steiner k-Wiener index SWk(G) of a connected graph G is defined as X SWk(G) = d(S): S⊆V (G) jSj=k 1 For k = 2, the Steiner k-Wiener index coincides with the Wiener index. The average k-Steiner n distance µk(G) is related to the Steiner k-Wiener index via the equality µk(G) = SWk(G)= k . In [15] the exact values of the Steiner k-Wiener index of the path, star, complete graph, and complete bipartite graph and sharp lower and upper bounds for SWk(G) for connected graphs and for trees have been obtained. In [12] an application of Steiner k-Wiener index in math- ematical chemistry is reported, and it is shown that the term W (G) + λSWk(G) provides a better approximation for the boiling points of alkanes than W (G) itself, and that the best such approximation is obtained for k = 7. For a survey on Steiner distance see [16]. The problem of deciding whether for a given subset of vertices in a graph G there exist a Steiner tree of size at most t belongs to the classical NP-complete problems from [11]. Hence for k > 2 it is not very likely to find an efficient way to compute the Steiner k-Wiener index for general graphs. Therefore it becomes interesting to either find efficient procedures to compute the Steiner k-Wiener index or bounds for particular classes of graphs. A vertex v is a cut vertex of graph G if deleting v and all edges incident to it increases the number of connected components G. A block of a graph is a maximal connected vertex induced subgraph that has no cut vertices. A block graph is a graph in which every block is a clique. Block graphs are a natural generalization of trees, and they arise in areas such as metric graph theory, [1], molecular graphs [3] and phylogenetics [10]. They have been characterized in various ways, for example, as certain intersection graphs [13], or in terms of distance conditions [3]. In [2] it has been shown that the determinant of the distance matrix of a block graph depends only on types of blocks, and not how they are connected. Steiner distance, Steiner centers and Steiner medians of block graphs have been studied in [19]. A vertex of a block graph G that appears in only one block is called a pendant vertex. Hence there are exacly two types of vertices in a block graph: cut vertices and pendant vertices. A block B is called a pendant block if B has a non-empty intersection with a unique block in G. Any block graph different from a complete graph has at least two pendant blocks. Pendant vertex from a pendant block of G is called a leaf of G. For a block graph G with blocks B1;B2;:::;Bt, let bi = jV (Bi)j, for i 2 f1; : : : ; tg. We call a sequence b1 ≥ b2 ≥ ::: ≥ bt the block order sequence of G. A line graph L(G) of a simple graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge if and only if the corresponding edges of G have a vertex in common. A graph is a line graph of a tree if it is a connected block graph in which each cut vertex is in exactly two blocks, hence they are claw-free block graphs: no induced subgraph is a claw - a complete bipartite graph K1;3: A caterpillar is a tree with the property that a path remains if all leaves are deleted. This path is called the backbone of the caterpillar. Line graphs of caterpillars are called path-like block graphs. These are all block graphs with precisely two pendant blocks. The windmill graph W d(r; t) is a block graph constructed by joining t copies of Kr at a shared vertex, where r; t ≥ 2. If v 2 V (G) is adjacent to all other vertices of G, it is called a universal vertex. A block graph with a universal vertex is called star-like block graph. For v 2 V (G) let N(v) denote the set of all neighbours. The degree of vertex v 2 V (G) in graph G is defined as the number of neighbours and denoted with deg(v), i.e., deg(v) = jN(v)j. Graph G has degree sequence λ = (d1; : : : ; dn), d1 ≥ d2 ≥ ::: ≥ dk, if vertices of G can be indexed from v1 to vn such that deg(vi) = di, i = 1; : : : ; n. Let T (λ) be the set of trees with degree sequence λ. It is known that T (λ) is not empty, if and only if d1 + ::: + dn = 2(n − 1). The Cartesian product G H of two graphs G and H is the graph with vertex set V (G) × V (H) and (a; x)(b; y) 2 E(G H) whenever either ab 2 E(G) and x = y, or a = b and xy 2 E(H). Cartesian products of complete graphs are called Hamming graphs. They can be alternatively described as follows. For i = 1; 2; : : : ; t let ri ≥ 2 be given integers. Let G be the graph whose vertices are the t-tuples a1; a2; : : : at with ai 2 f0; 1; : : : ; ri − 1g. Two vertices being 2 adjacent if the corresponding tuples differ in precisely one place. Then it is straightforward to see that G is isomorphic to Kr1 2Kr2 2 ··· 2Krt . A subgraph H of G is called isometric (or distance preserving) if dH (u; v) = dG(u; v) for all u; v 2 V (H). Isometric subgraphs of Hamming graphs are called partial Hamming graphs. The Hamming distance between two t-tuples is defined as the number of positions in which these t-tuples differ, in a partial Hamming graph it coincides with the geodetic distance. In this paper we obtain several simple methods for calculating the Steiner k-Wiener index of block graphs. In Section2 we present the block decomposition formula of the Steiner k- Wiener index of block graphs. In Section3 we present the edge decomposition formula of the Steiner 3-Wiener index of block graphs. In Section4 we present the vertex decomposition formula of the Steiner k-Wiener index of block graphs and relate it to the k-Steiner betweenness centrality. In Section5 we study the graphs which minimize or maximize Steiner k-Wiener index among all block graphs with the same set of blocks, and obtain the sharp lower bound. To describe block graphs that maximize the Steiner k-Wiener index we introduce a special graph transformation called generalized block shift, which generalizes the generalized tree shift transformation introduced in [8] by Csikv´ari. 2 Block decomposition formula of Steiner k-Wiener index of block graphs Let n(G) denote the number of vertices of a graph G.
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