Audace Amen Vioutou DOSSOU-OLORY | Université d'Abomey_calavi (UAC) (original) (raw)

Papers by Audace Amen Vioutou DOSSOU-OLORY

arXiv (Cornell University), May 16, 2020

The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities... more The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of G is the sum of the distances between all unordered pairs of vertices of G. The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the Harary index, the generalised Wiener index W λ for λ > 0 and λ < 0, and the reciprocal complementary Wiener index. Our results imply and unify known bounds on these Wiener-type indices for trees of given order and diameter. We also present similar results for the k-Steiner Wiener index of trees with a given eccentric sequence. The Steiner distance of a set A ⊆ V (G) is the minimum number of edges in a subtree of G whose vertex set contains A, and the k-Steiner Wiener index is the sum of distances of all k-element subsets of V (G). As a corollary, we obtain a sharp lower bound on the k-Steiner Wiener index of trees with given order and diameter, and determine in which cases the extremal tree is unique, thereby correcting an error in the literature.

arXiv (Cornell University), Oct 8, 2019

A vertex whose removal in a graph G increases the number of components of G is called a cut verte... more A vertex whose removal in a graph G increases the number of components of G is called a cut vertex. For all n, c, we determine the maximum number of connected induced subgraphs in a connected graph with order n and c cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case c > 0 remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the corresponding formulas for the number of connected induced subgraphs. The 'minimal' graph in this case is a tree, thus coincides with the structure that was given by Li and Wang [Further analysis on the total number of subtrees of trees.

Australas. J Comb., 2021

A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D ... more A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D inside a larger tree T is the proportion of such leaf-induced subtrees in T that are isomorphic to D among all those with the same number of leaves as D. The inducibility of D measures how large this density can be as the size of T tends to infinity. In this paper, we explicitly determine the inducibility in some previously unknown cases and find general upper and lower bounds, in particular in the case where D is balanced, i.e., when its branches have at least almost the same size. Moreover, we prove a result on the speed of convergence of the maximum density of D in strictly d-ary trees T (trees where every internal vertex has precisely d children) of a given size n to the inducibility as n → ∞, which supports an open conjecture.

Discrete Mathematics & Theoretical Computer Science, Oct 17, 2019

The quantity that captures the asymptotic value of the maximum number of appearances of a given t... more The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree 1) S with k leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of S. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Székely, and the second author of this article, we provide bounds for the inducibility J(A5) of the 5-leaf binary tree A5 whose branches are a single leaf and the complete binary tree of height 2. It was indicated before that J(A5) appears to be 'close' to 1/4. We can make this precise by showing that 0.24707. .. ≤ J(A5) ≤ 0.24745. . .. Furthermore, we also consider the problem of determining the inducibility of the tree Q4, which is the only tree among 4-leaf topological trees for which the inducibility is unknown.

arXiv (Cornell University), Mar 3, 2023

We characterise the structure of those graphs of a given order which maximise the number of conne... more We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number, vertex cover number, vertex connectivity, edge connectivity, chromatic number, number of bridges, thereby contributing to filling a gap in the literature.

arXiv (Cornell University), Feb 9, 2020

Cut vertices are often used as a measure of nodes' importance within a network. They are those no... more Cut vertices are often used as a measure of nodes' importance within a network. They are those nodes whose failure disconnects a graph. Let N(G) be the number of connected induced subgraphs of a graph G. In this work, we investigate the maximum of N(G) where G is a unicyclic graph with n nodes of which c are cut vertices. For all valid n, c, we give a full description of those maximal (that maximise N(.)) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of n, c, however, there is a unique maximal unicyclic graph with n nodes and c cut vertices. In particular, the well-known negative correlation between the number of connected induced subgraphs of trees and the Wiener index (sum of distances) fails for unicyclic graphs with n nodes and c cut vertices: for instance, the maximal unicyclic graph with n = 3, 4 mod 5 nodes and c = n − 5 > 3 cut vertices is different from the unique graph that was shown by Tan et al. [The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices. J. Appl. Math. Comput., 55:1-24, 2017] to minimise the Wiener index. Our main characterisation of maximal unicyclic graphs with respect to the number of connected induced subgraphs also applies to unicyclic graphs with n nodes, c cut vertices and girth at most g > 3, since it is shown that the girth of every maximal graph with n nodes and c cut vertices cannot exceed 4.

Discrete Mathematics, Feb 1, 2020

Imitating the binary inducibility, a recently introduced invariant of binary trees (Czabarka et a... more Imitating the binary inducibility, a recently introduced invariant of binary trees (Czabarka et al., 2017), we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2). We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T | −1/2) compared to the general case where it is shown that the difference is O(|T | −1) which, in particular, responds positively to a conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.

arXiv (Cornell University), Feb 19, 2018

Trees without vertices of degree 2 are sometimes named topological trees. In this work, we bring ... more Trees without vertices of degree 2 are sometimes named topological trees. In this work, we bring forward the study of the inducibility of (rooted) topological trees with a given number of leaves. The inducibility of a topological tree S is the limit superior of the proportion of all subsets of leaves of T that induce a copy of S as the size of T grows to infinity. In particular, this relaxes the degree-restriction for the existing notion of the inducibility in d-ary trees. We discuss some of the properties of this generalised concept and investigate its connection with the degree-restricted inducibility. In addition, we prove that stars and binary caterpillars are the only topological trees that have an inducibility of 1. We also find an explicit lower bound on the limit inferior of the proportion of all subsets of leaves of T that induce either a star or a binary caterpillar as the size of T tends to infinity.

arXiv (Cornell University), Apr 16, 2018

Given d ≥ 2 and two rooted d-ary trees D and T such that D has k leaves, the density γ(D, T) of D... more Given d ≥ 2 and two rooted d-ary trees D and T such that D has k leaves, the density γ(D, T) of D in T is the proportion of all k-element subsets of leaves of T that induce a tree isomorphic to D, after erasing all vertices of outdegree 1. In a recent work, it was proved that the limit inferior of this density as the size of T grows to infinity is always zero unless D is the k-leaf binary caterpillar F 2 k (the binary tree with the property that a path remains upon removal of all the k leaves). Our main theorem in this paper is an exact formula (involving both d and k) for the limit inferior of γ(F 2 k , T) as the size of T tends to infinity.

arXiv (Cornell University), Feb 11, 2018

Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of ... more Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T | −1/2) compared to the general case where it is shown that the difference is O(|T | −1) which, in particular, responds positively to an existing conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.

arXiv (Cornell University), Feb 11, 2018

Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of ... more Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T | −1/2) compared to the general case where it is shown that the difference is O(|T | −1) which, in particular, responds positively to an existing conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.

arXiv (Cornell University), Nov 29, 2018

The quantity that captures the asymptotic value of the maximum number of appearances of a given t... more The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree 1) S with k leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of S. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Székely, and the second author of this article, we provide bounds for the inducibility J(A5) of the 5-leaf binary tree A5 whose branches are a single leaf and the complete binary tree of height 2. It was indicated before that J(A5) appears to be 'close' to 1/4. We can make this precise by showing that 0.24707. .. ≤ J(A5) ≤ 0.24745. . .. Furthermore, we also consider the problem of determining the inducibility of the tree Q4, which is the only tree among 4-leaf topological trees for which the inducibility is unknown.

Acta Applicandae Mathematicae, Jan 13, 2021

The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities... more The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of G is the sum of the distances between all unordered pairs of vertices of G. The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the Harary index, the generalised Wiener index W λ for λ > 0 and λ < 0, and the reciprocal complementary Wiener index. Our results imply and unify known bounds on these Wiener-type indices for trees of given order and diameter. We also present similar results for the k-Steiner Wiener index of trees with a given eccentric sequence. The Steiner distance of a set A ⊆ V (G) is the minimum number of edges in a subtree of G whose vertex set contains A, and the k-Steiner Wiener index is the sum of distances of all k-element subsets of V (G). As a corollary, we obtain a sharp lower bound on the k-Steiner Wiener index of trees with given order and diameter, and determine in which cases the extremal tree is unique, thereby correcting an error in the literature.

arXiv (Cornell University), Dec 6, 2018

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimis... more Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of connected induced subgraphs coincide with those that are known to maximise the Wiener index (the sum of the distances between all unordered pairs of vertices), and vice versa. For every k, we also determine the connected graphs that are extremal with respect to the number of kvertex connected induced subgraphs. We show that, in contrast to the minimum which is uniquely realised by the path, the maximum value is attained by a rich class of connected graphs.

arXiv (Cornell University), Feb 17, 2020

The eccentricity of a vertex u in a connected graph G is the distance between u and a vertex fart... more The eccentricity of a vertex u in a connected graph G is the distance between u and a vertex farthest from it; the eccentric sequence of G is the nondecreasing sequence of the eccentricities of G. In this paper, we determine the unique tree that minimises the Wiener index, i.e. the sum of distances between all unordered vertex pairs, among all trees with a given eccentric sequence. We show that the same tree maximises the number of subtrees among all trees with a given eccentric sequence, thus providing another example of negative correlation between the number of subtrees and the Wiener index of trees. Furthermore, we provide formulas for the corresponding extreme values of these two invariants in terms of the eccentric sequence. As a corollary to our results, we determine the unique tree that minimises the edge Wiener index, the vertex-edge Wiener index, the Schulz index (or degree distance), and the Gutman index among all trees with a given eccentric sequence.

arXiv (Cornell University), Jun 26, 2022

A subtree can be induced in a natural way by a subset of leaves of a rooted tree. We study the nu... more A subtree can be induced in a natural way by a subset of leaves of a rooted tree. We study the number of nonisomorphic such subtrees induced by leaves (leaf-induced subtrees) of a rooted tree with no vertex of outdegree 1 (topological tree). We show that only stars and binary caterpillars have the minimum nonisomorphic leaf-induced subtrees among all topological trees with a given number of leaves. We obtain a closed formula and a recursive formula for the families of d-ary caterpillars and complete d-ary trees, respectively. An asymptotic formula is found for complete d-ary trees using polynomial recurrences. We also show that the complete binary tree of height h > 1 contains precisely 2(1.24602...) 2 h nonisomorphic leaf-induced subtrees.

DOAJ (DOAJ: Directory of Open Access Journals), 2019

In analogy to a concept of Fibonacci trees, we define the leaf-Fibonacci tree of size n and inves... more In analogy to a concept of Fibonacci trees, we define the leaf-Fibonacci tree of size n and investigate its number of nonisomorphic leaf-induced subtrees. Denote by f 0 the one vertex tree and f 1 the tree that consists of a root with two leaves attached to it; the leaf-Fibonacci tree f n of size n ≥ 2 is the binary tree whose branches are f n−1 and f n−2. We derive a nonlinear difference equation for the number N(f n) of nonisomorphic leaf-induced subtrees (subtrees induced by leaves) of f n , and also prove that N(f n) is asymptotic to 1.00001887227319. .. (1.48369689570172. . .) φ n (φ = golden ratio) as n grows to infinity.

European journal of mathematics, Jan 11, 2021

A vertex whose removal in a graph G increases the number of components of G is called a cut verte... more A vertex whose removal in a graph G increases the number of components of G is called a cut vertex. For all n, c, we determine the maximum number of connected induced subgraphs in a connected graph with order n and c cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case c > 0 remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the corresponding formulas for the number of connected induced subgraphs. The 'minimal' graph in this case is a tree, thus coincides with the structure that was given by Li and Wang [Further analysis on the total number of subtrees of trees.

arXiv (Cornell University), Nov 27, 2018

A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D ... more A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D inside a larger tree T is the proportion of such leaf-induced subtrees in T that are isomorphic to D among all those with the same number of leaves as D. The inducibility of D measures how large this density can be as the size of T tends to infinity. In this paper, we explicitly determine the inducibility in some previously unknown cases and find general upper and lower bounds, in particular in the case where D is balanced, i.e., when its branches have at least almost the same size. Moreover, we prove a result on the speed of convergence of the maximum density of D in strictly d-ary trees T (trees where every internal vertex has precisely d children) of a given size n to the inducibility as n → ∞, which supports an open conjecture.

Le Centre pour la Communication Scientifique Directe - HAL - Inria, Jan 22, 2020

Denote by G(n, d, g, k) the set of all connected graphs of order n, having d > 0 cycles, girth g ... more Denote by G(n, d, g, k) the set of all connected graphs of order n, having d > 0 cycles, girth g and k pendent vertices. In this paper, we give a partial characterisation of the structure of all maximal graphs in G(n, d, g, k) for the number of connected induced subgraphs. For the special case d = 1, we find a complete characterisation of all maximal unicyclic graphs. We also derive a precise formula for the maximum number of connected induced subgraphs given: (1) order, girth, and number of pendent vertices; (2) order and girth; (3) order.

arXiv (Cornell University), May 16, 2020

The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities... more The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of G is the sum of the distances between all unordered pairs of vertices of G. The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the Harary index, the generalised Wiener index W λ for λ > 0 and λ < 0, and the reciprocal complementary Wiener index. Our results imply and unify known bounds on these Wiener-type indices for trees of given order and diameter. We also present similar results for the k-Steiner Wiener index of trees with a given eccentric sequence. The Steiner distance of a set A ⊆ V (G) is the minimum number of edges in a subtree of G whose vertex set contains A, and the k-Steiner Wiener index is the sum of distances of all k-element subsets of V (G). As a corollary, we obtain a sharp lower bound on the k-Steiner Wiener index of trees with given order and diameter, and determine in which cases the extremal tree is unique, thereby correcting an error in the literature.

arXiv (Cornell University), Oct 8, 2019

A vertex whose removal in a graph G increases the number of components of G is called a cut verte... more A vertex whose removal in a graph G increases the number of components of G is called a cut vertex. For all n, c, we determine the maximum number of connected induced subgraphs in a connected graph with order n and c cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case c > 0 remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the corresponding formulas for the number of connected induced subgraphs. The 'minimal' graph in this case is a tree, thus coincides with the structure that was given by Li and Wang [Further analysis on the total number of subtrees of trees.

Australas. J Comb., 2021

A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D ... more A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D inside a larger tree T is the proportion of such leaf-induced subtrees in T that are isomorphic to D among all those with the same number of leaves as D. The inducibility of D measures how large this density can be as the size of T tends to infinity. In this paper, we explicitly determine the inducibility in some previously unknown cases and find general upper and lower bounds, in particular in the case where D is balanced, i.e., when its branches have at least almost the same size. Moreover, we prove a result on the speed of convergence of the maximum density of D in strictly d-ary trees T (trees where every internal vertex has precisely d children) of a given size n to the inducibility as n → ∞, which supports an open conjecture.

Discrete Mathematics & Theoretical Computer Science, Oct 17, 2019

The quantity that captures the asymptotic value of the maximum number of appearances of a given t... more The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree 1) S with k leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of S. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Székely, and the second author of this article, we provide bounds for the inducibility J(A5) of the 5-leaf binary tree A5 whose branches are a single leaf and the complete binary tree of height 2. It was indicated before that J(A5) appears to be 'close' to 1/4. We can make this precise by showing that 0.24707. .. ≤ J(A5) ≤ 0.24745. . .. Furthermore, we also consider the problem of determining the inducibility of the tree Q4, which is the only tree among 4-leaf topological trees for which the inducibility is unknown.

arXiv (Cornell University), Mar 3, 2023

We characterise the structure of those graphs of a given order which maximise the number of conne... more We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number, vertex cover number, vertex connectivity, edge connectivity, chromatic number, number of bridges, thereby contributing to filling a gap in the literature.

arXiv (Cornell University), Feb 9, 2020

Cut vertices are often used as a measure of nodes' importance within a network. They are those no... more Cut vertices are often used as a measure of nodes' importance within a network. They are those nodes whose failure disconnects a graph. Let N(G) be the number of connected induced subgraphs of a graph G. In this work, we investigate the maximum of N(G) where G is a unicyclic graph with n nodes of which c are cut vertices. For all valid n, c, we give a full description of those maximal (that maximise N(.)) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of n, c, however, there is a unique maximal unicyclic graph with n nodes and c cut vertices. In particular, the well-known negative correlation between the number of connected induced subgraphs of trees and the Wiener index (sum of distances) fails for unicyclic graphs with n nodes and c cut vertices: for instance, the maximal unicyclic graph with n = 3, 4 mod 5 nodes and c = n − 5 > 3 cut vertices is different from the unique graph that was shown by Tan et al. [The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices. J. Appl. Math. Comput., 55:1-24, 2017] to minimise the Wiener index. Our main characterisation of maximal unicyclic graphs with respect to the number of connected induced subgraphs also applies to unicyclic graphs with n nodes, c cut vertices and girth at most g > 3, since it is shown that the girth of every maximal graph with n nodes and c cut vertices cannot exceed 4.

Discrete Mathematics, Feb 1, 2020

Imitating the binary inducibility, a recently introduced invariant of binary trees (Czabarka et a... more Imitating the binary inducibility, a recently introduced invariant of binary trees (Czabarka et al., 2017), we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2). We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T | −1/2) compared to the general case where it is shown that the difference is O(|T | −1) which, in particular, responds positively to a conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.

arXiv (Cornell University), Feb 19, 2018

Trees without vertices of degree 2 are sometimes named topological trees. In this work, we bring ... more Trees without vertices of degree 2 are sometimes named topological trees. In this work, we bring forward the study of the inducibility of (rooted) topological trees with a given number of leaves. The inducibility of a topological tree S is the limit superior of the proportion of all subsets of leaves of T that induce a copy of S as the size of T grows to infinity. In particular, this relaxes the degree-restriction for the existing notion of the inducibility in d-ary trees. We discuss some of the properties of this generalised concept and investigate its connection with the degree-restricted inducibility. In addition, we prove that stars and binary caterpillars are the only topological trees that have an inducibility of 1. We also find an explicit lower bound on the limit inferior of the proportion of all subsets of leaves of T that induce either a star or a binary caterpillar as the size of T tends to infinity.

arXiv (Cornell University), Apr 16, 2018

Given d ≥ 2 and two rooted d-ary trees D and T such that D has k leaves, the density γ(D, T) of D... more Given d ≥ 2 and two rooted d-ary trees D and T such that D has k leaves, the density γ(D, T) of D in T is the proportion of all k-element subsets of leaves of T that induce a tree isomorphic to D, after erasing all vertices of outdegree 1. In a recent work, it was proved that the limit inferior of this density as the size of T grows to infinity is always zero unless D is the k-leaf binary caterpillar F 2 k (the binary tree with the property that a path remains upon removal of all the k leaves). Our main theorem in this paper is an exact formula (involving both d and k) for the limit inferior of γ(F 2 k , T) as the size of T tends to infinity.

arXiv (Cornell University), Feb 11, 2018

Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of ... more Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T | −1/2) compared to the general case where it is shown that the difference is O(|T | −1) which, in particular, responds positively to an existing conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.

arXiv (Cornell University), Feb 11, 2018

Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of ... more Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T | −1/2) compared to the general case where it is shown that the difference is O(|T | −1) which, in particular, responds positively to an existing conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.

arXiv (Cornell University), Nov 29, 2018

The quantity that captures the asymptotic value of the maximum number of appearances of a given t... more The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree 1) S with k leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of S. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Székely, and the second author of this article, we provide bounds for the inducibility J(A5) of the 5-leaf binary tree A5 whose branches are a single leaf and the complete binary tree of height 2. It was indicated before that J(A5) appears to be 'close' to 1/4. We can make this precise by showing that 0.24707. .. ≤ J(A5) ≤ 0.24745. . .. Furthermore, we also consider the problem of determining the inducibility of the tree Q4, which is the only tree among 4-leaf topological trees for which the inducibility is unknown.

Acta Applicandae Mathematicae, Jan 13, 2021

The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities... more The eccentric sequence of a connected graph G is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of G is the sum of the distances between all unordered pairs of vertices of G. The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the Harary index, the generalised Wiener index W λ for λ > 0 and λ < 0, and the reciprocal complementary Wiener index. Our results imply and unify known bounds on these Wiener-type indices for trees of given order and diameter. We also present similar results for the k-Steiner Wiener index of trees with a given eccentric sequence. The Steiner distance of a set A ⊆ V (G) is the minimum number of edges in a subtree of G whose vertex set contains A, and the k-Steiner Wiener index is the sum of distances of all k-element subsets of V (G). As a corollary, we obtain a sharp lower bound on the k-Steiner Wiener index of trees with given order and diameter, and determine in which cases the extremal tree is unique, thereby correcting an error in the literature.

arXiv (Cornell University), Dec 6, 2018

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimis... more Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of connected induced subgraphs coincide with those that are known to maximise the Wiener index (the sum of the distances between all unordered pairs of vertices), and vice versa. For every k, we also determine the connected graphs that are extremal with respect to the number of kvertex connected induced subgraphs. We show that, in contrast to the minimum which is uniquely realised by the path, the maximum value is attained by a rich class of connected graphs.

arXiv (Cornell University), Feb 17, 2020

The eccentricity of a vertex u in a connected graph G is the distance between u and a vertex fart... more The eccentricity of a vertex u in a connected graph G is the distance between u and a vertex farthest from it; the eccentric sequence of G is the nondecreasing sequence of the eccentricities of G. In this paper, we determine the unique tree that minimises the Wiener index, i.e. the sum of distances between all unordered vertex pairs, among all trees with a given eccentric sequence. We show that the same tree maximises the number of subtrees among all trees with a given eccentric sequence, thus providing another example of negative correlation between the number of subtrees and the Wiener index of trees. Furthermore, we provide formulas for the corresponding extreme values of these two invariants in terms of the eccentric sequence. As a corollary to our results, we determine the unique tree that minimises the edge Wiener index, the vertex-edge Wiener index, the Schulz index (or degree distance), and the Gutman index among all trees with a given eccentric sequence.

arXiv (Cornell University), Jun 26, 2022

A subtree can be induced in a natural way by a subset of leaves of a rooted tree. We study the nu... more A subtree can be induced in a natural way by a subset of leaves of a rooted tree. We study the number of nonisomorphic such subtrees induced by leaves (leaf-induced subtrees) of a rooted tree with no vertex of outdegree 1 (topological tree). We show that only stars and binary caterpillars have the minimum nonisomorphic leaf-induced subtrees among all topological trees with a given number of leaves. We obtain a closed formula and a recursive formula for the families of d-ary caterpillars and complete d-ary trees, respectively. An asymptotic formula is found for complete d-ary trees using polynomial recurrences. We also show that the complete binary tree of height h > 1 contains precisely 2(1.24602...) 2 h nonisomorphic leaf-induced subtrees.

DOAJ (DOAJ: Directory of Open Access Journals), 2019

In analogy to a concept of Fibonacci trees, we define the leaf-Fibonacci tree of size n and inves... more In analogy to a concept of Fibonacci trees, we define the leaf-Fibonacci tree of size n and investigate its number of nonisomorphic leaf-induced subtrees. Denote by f 0 the one vertex tree and f 1 the tree that consists of a root with two leaves attached to it; the leaf-Fibonacci tree f n of size n ≥ 2 is the binary tree whose branches are f n−1 and f n−2. We derive a nonlinear difference equation for the number N(f n) of nonisomorphic leaf-induced subtrees (subtrees induced by leaves) of f n , and also prove that N(f n) is asymptotic to 1.00001887227319. .. (1.48369689570172. . .) φ n (φ = golden ratio) as n grows to infinity.

European journal of mathematics, Jan 11, 2021

A vertex whose removal in a graph G increases the number of components of G is called a cut verte... more A vertex whose removal in a graph G increases the number of components of G is called a cut vertex. For all n, c, we determine the maximum number of connected induced subgraphs in a connected graph with order n and c cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case c > 0 remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the corresponding formulas for the number of connected induced subgraphs. The 'minimal' graph in this case is a tree, thus coincides with the structure that was given by Li and Wang [Further analysis on the total number of subtrees of trees.

arXiv (Cornell University), Nov 27, 2018

A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D ... more A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D inside a larger tree T is the proportion of such leaf-induced subtrees in T that are isomorphic to D among all those with the same number of leaves as D. The inducibility of D measures how large this density can be as the size of T tends to infinity. In this paper, we explicitly determine the inducibility in some previously unknown cases and find general upper and lower bounds, in particular in the case where D is balanced, i.e., when its branches have at least almost the same size. Moreover, we prove a result on the speed of convergence of the maximum density of D in strictly d-ary trees T (trees where every internal vertex has precisely d children) of a given size n to the inducibility as n → ∞, which supports an open conjecture.

Le Centre pour la Communication Scientifique Directe - HAL - Inria, Jan 22, 2020

Denote by G(n, d, g, k) the set of all connected graphs of order n, having d > 0 cycles, girth g ... more Denote by G(n, d, g, k) the set of all connected graphs of order n, having d > 0 cycles, girth g and k pendent vertices. In this paper, we give a partial characterisation of the structure of all maximal graphs in G(n, d, g, k) for the number of connected induced subgraphs. For the special case d = 1, we find a complete characterisation of all maximal unicyclic graphs. We also derive a precise formula for the maximum number of connected induced subgraphs given: (1) order, girth, and number of pendent vertices; (2) order and girth; (3) order.