Path-tables of trees: a survey and some new results

The (vertex) path-table of a tree T contains quantitative information about the paths in T . The entry (i, j) of this table gives the number of paths of length j passing through vertex vi. The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in general, any type of path-table of a tree T does not determine T uniquely. We shall show that in trees, the number of paths passing through edge xy can only be expressed in terms of paths passing through vertices x and y up to a length of 4. In contrast to the vertex path-table, we show that the row of the edge path-table corresponding to the central edge of a tree T of odd diameter, is unique in the table. Finally we show that special classes of trees such as caterpillars and restricted thin trees (RTT) are reconstructible from their path-tables. 1 Historical background on tables involving number of paths M.Randić has been a major contributor to the development of mathematical chemistry. In particular he wanted to give the concept of atomic paths a formal mathematical development based on the use of Graph Theory. In [19], he discussed in some detail the uses of the enumeration of paths and neighbours in molecular graphs and tried to characterise molecular graphs through the use of the atomic path code of a molecule. Randić conjectured that the list of path numbers determines the graph G uniquely and verified this assertion for graphs up to 11 vertices. Later Bloom et al. [2] defined di,j as the number of vertices in a graph G of diameter d(G) that are at a distance j from vertex vi. The sequence (di,0, di,1 . . . di,j . . . di,d(G)) is then called the distance degree sequence of vi in G. The ∣G∣-tuple of distance degree sequences of the vertices of G with entries arranged in lexicographic order is called the Distance Degree Sequence of G (DDS(G)). Similarly they also defined the path degree sequence of vi in G as the sequence (pi,0, pi,1 . . . pi,j . . . pi,d(G)) where pi,j is the number of paths in G starting at vertex vi and having length j. The ordered set of all such sequences arranged in lexicographic order is called the Path Degree Sequence of G (PDS(G)). In his study, Randić remarked that since there is a unique path between pairs of atoms in acyclic structures, the number of paths of a given length corresponds to the number of neighbours at a given distance. Quintas and Slater [18] used Randić’s remark in order to prove that for a connected graph G, DDS(G) = PDS(G) if and only if G is a tree. They also pointed out that when G is thought of as a molecular graph, PDS(G) is precisely the lexicographically ordered list of atomic codes for the atoms (vertices) of G. Slater [24] first showed that Randić’s conjecture is not valid by constructing pairs of non-isomorphic trees having the same path degree sequence and then together with Quintas [18] they constructed also non-tree graphs having a variety of properties and which also invalidate the conjecture. Since a tree is not, in general, characterised by its Path(Distance) Degree Sequence, researchers turned their focus on finding the least possible order for the existence of pairs of non-isomorphic graphs having the same PDS. They found out that the least order for which there exists a pair of non-isomorphic trees with the same PDS is greater than 15 1 Asciak: Path-tables of trees Published by Digital Commons@Georgia Southern, 2015 for trees with no vertices of degree greater than 4 and less than 19 for trees without any vertex degree restrictions [18]. V. A. Skorobogatov and A.A.Dobrynin [23] introduced the path layer matrix, a matrix characterising the paths that can be found in a graph. The path layer matrix of a graph G of order p is the p × (p − 1) matrix ¿(G) = ∣∣¿i,j∣∣, where ¿i,j is the number of simple paths in G starting at vertex vi and having length j. By ordering the rows of ¿(G) in decreasing lengths ( “length” here being the number of nonzero elements) and then by lexicographically arranging the rows with the same length, one can obtain a canonical form of ¿(G). This path layer matrix is also known as the path degree sequence of a graph or the atomic path code of a molecule. This invariant and its modifications have found some important applications in chemistry especially in the characterisation of branching in molecules, establishing similarity of molecular graphs and for drug design [20, 21, 22] As pointed out earlier, the path layer matrix of graph G having a sufficiently large order, does not characterise every graph G uniquely. The problem of bounding this order has been studied for over twenty years. Dobrynin and Mel’nikov [9] noticed that mathematical investigations of this matrix deal with finding a pair of non-isomorphic graphs having some specific properties and such that both graphs have the same path layer matrix. Among these properties we can have the girth, cyclomatic number and planarity of graphs [6, 9, 14, 15, 16]. Thus an interesting problem is to determine the least possible order of a pair of non-isomorphic graphs in such a class with the same path layer matrix. In fact in [18], Quintas and Slater proposed the following problem : Does there exist a pair of connected non-isomorphic r-regular graphs (graphs in which every vertex has degree r) having the same path degree sequences? If the answer is yes, then for each r ≥ 3, what is the least order p(r) possible for graphs in such a pair? Defining f1 to be the minimal order such that there exist non-isomorphic graphs of order f1 having the same path layer matrix, Dobrynin [4] and Randić [19] showed that 12 ≤ f1 ≤ 14 for general graphs. Dobrynin restricted the problem to certain sub-classes of graphs by first considering r-regular graphs and then r-regular graphs without cutvertices (a cut-vertex is one whose removal disconnects the graph). Balaban et al. were the first researchers who found a pair of cubic graphs of order 142 that have the same path layer matrix [1]. Then Dobrynin [5] showed that for every r ≥ 3, r-regular graphs with the same path layer matrix can be constructed and the least order for these pairs of graphs is a linear function of r when r ≥ 5 while for cubic graphs f1 ≤ 116 and for 4-regular graphs f1 ≤ 114. In [7] the upperbound for the order of cubic graphs has been improved to 62. More recently it has been discovered that if p(r) is the least order of pairs of nonisomorphic r-regular graphs having the same path layer matrix then 20 ≤ p(3) ≤ 36; [8] 16 ≤ p(4) ≤ 18; [25] 12 ≤ p(5) ≤ 48; [26] 12 ≤ p(6) ≤ 51; [26] Now since the key feature of all graphs with same path layer matrix was that they 2 Theory and Applications of Graphs, Vol. 2 [2015], Iss. 2, Art. 4 http://digitalcommons.georgiasouthern.edu/tag/vol2/iss2/4 DOI: 10.20429/tag.2015.020204 contained cut-vertices then this prompted the investigation of finding a pair of nonisomorphic r-regular graphs without cut-vertices and having the same path layer matrix. Defining f2 to be the the least order for pairs of non-isomorphic graphs without cutvertices and having the same path layer matrix, Dobrynin [8] proved that for cubic graphs f2 ≤ 31 and Yuansheng et al. [25] discovered a construction for a pair of nonisomorphic 4-regular graphs having the same path layer matrix and proved that f2 ≤ 18. Recently H.Chung [3] constructed a pair of non-isomorphic 5-regular graphs without cut-vertices of order 20 having the same path layer matrix in which f2 is reduced from 48 to 20. After proving a converse of Kelly’s Lemma in graph reconstruction, Dulio and Pannone [10] defined a slightly different kind of table which they call path-table. They stated that there must be a class of graphs Ϝ in between the empty class and the class of all graphs of order less than n with the following property. Let X and Y be trees of order n in which there is a bijection f : V (X) −→ V (Y ) such that, if for every graph Q in Ϝ, the number of subgraphs of X containing a vertex x of V (X) and isomorphic to Q is the same as the number of subgraphs of Y containing f(x) and isomorphic to Q, then X ≃ Y . There can be many such classes Ϝ but the first and most natural class to study for trees is the class Ϝ of all paths. Thus the statement to be tested becomes: If X and Y are trees of order n and there is a bijection f : V (X) −→ V (Y ) such that, for every path in Ϝ, the number of paths of X containing a vertex x of V (X) of length l is the same as the number of paths of Y containing f(x) of length l, then X ≃ Y . The assumption in the above statement is tantamount to saying that X and Y have the same path-table (up to re-ordering of the rows). Thus rather than considering paths starting at a vertex, they considered paths passing through a vertex (that is, containing a given vertex, possibly as its endvertex). Like Slater they also showed that trees having the same path-tables need not be isomorphic. In [11], they proved that there exist infinitely many pairs of non-isomorphic trees having the same pathand path layer-tables. The smallest tree that can be obtained with their construction has twenty vertices. In this paper we shall take a closer look at these two path-tables for trees, and we shall also introduce an analogous table involving the number of paths passing through edges. We shall study possible relationships between these tables and we shall also try to identify classes of trees which are uniquely determined by some type of path-table. We shall review some known results and present some new ones. 2 Definitions We shall start by giving uniform definitions for the three path-tables which we shall be studying. Let T be a tree and let its vertices be v1, v2, . . . , vn and edges e1, e2, . . . , em. Let d = diam(T) be the length of the longest path in T . For any vi, let sT (vi) be the d-tuple whose j entry equals the number of paths of length j starting at vi and let pT (vi) be the d-tuple where j entry equals the number of paths of length j passing through vi. In both cases we shall drop the suffix T when the tree in question is clear from the context. We shall then denote the j entry of s(v) or p(v) by sj(v) and pj(v), respectively. 3 Asciak: Path-tables of trees Published by Digital Commons@Georgia Southern, 2015 Note that, in both cases, the first entry equals the degree of vi. The Randić table, which we shall here denote by S(T), is the n × d matrix whose i row is sT (vi). Dulio and Pannone’s table, which we shall here call the Vertex path-table of T and denoted by V P (T ) will be the n× d matrix where i row is pT (vi). Now, given an edge ei of T , let pT (ei) be the d-tuple whose j th entry equals the number of paths of length j passing through ei. The Edge path-table EP (T ) of T is the (n− 1)× d matrix whose i row is pT (ei). Again, we shall also drop the suffix T and denote the j entry of p(ei) by pj(ei). If ei is the edge xy we shall denote this by pj(x, y). We shall also write p(x, y) for p(ei). Clearly, for any given T , S(T ), V P (T ) and EP (T ) are unique up to re-ordering of its rows, and if two Randić tables or vertex path-tables or edge path-tables are such that one can be obtained from the other by a re-ordering of its rows, we shall consider the two tables to be the same. 3 A Path-table does not determine its tree, in general It is not a priori evident that the “passing” vector pT (v) of a vertex v contains more information than the “starting” vector sT (v). By definition, the first component is deg(v) for both and the i component of the former contains the i component of the latter as a summand. But one can verify that the example pairs of non-isomorphic trees T1, T2 with S(T1) = S(T2) provided by Slater [24] have V P (T1) ∕= V P (T2). However, in [11], Dulio and Pannone provided other families of pairs of non-isomorphic trees U1 and U2 with the same vertex path-table. These considerations give rise to the interesting open problems of classifying nonisomorphic trees that share a given vertex path-table and of identifying classes of trees which are uniquely determined by their path-tables. The main aim of this paper is to survey the known results in the area and to present some new ones of our own. Figure 1 shows a pair of Slater’s counterexamples on 18 vertices showing non-isomorphic trees with the same Randić-table. Table 1 shows their Randić-table but Table 2 shows their V P tables which are different. 4 Theory and Applications of Graphs, Vol. 2 [2015], Iss. 2, Art. 4 http://digitalcommons.georgiasouthern.edu/tag/vol2/iss2/4 DOI: 10.20429/tag.2015.020204