Darko Dimitrov - Academia.edu (original) (raw)

Papers by Darko Dimitrov

Symmetry detection is an important problem with many applications in pattern recognition, compute... more Symmetry detection is an important problem with many applications in pattern recognition, computer vision and computational geometry. In this paper, we propose a novel algorithm for computing a hyperplane of reflexive symmetry of a point set in arbitrary dimension with approximate symmetry. The algorithm is based on the geometric hashing technique. In addition, we consider a relation between the perfect reflective symmetry and the principal components of shapes, a relation that was already a base of few heuristic approaches that tackle the symmetry problem in 2D and 3D. From mechanics, it is known that, if H is a plane of reflective symmetry of the 3D rigid body, then a principal component of the body is orthogonal to H. Here we extend that result to any point set (continuous or discrete) in arbitrary dimension.

Discrete Applied Mathematics, Sep 1, 2023

The problem of characterizing trees with minimal atom-bond-connectivity index (minimal-ABC trees)... more The problem of characterizing trees with minimal atom-bond-connectivity index (minimal-ABC trees) has a reputation as one of the most demanding recent open optimization problems in mathematical chemistry. Here firstly, we give an affirmative answer to the conjecture, which states that enough large minimal-ABC trees are comprised solely of a root vertex and so-called Dz-and Dz+1-branches. Based on the presented theoretical results here and some already known results, we obtain enough constraints to reduce the search space and solve the optimization problem, and thus, determine exactly the minimal-ABC trees of a given arbitrary order.

Mathematical Inequalities & Applications, 2023

For a graph G of order n , size m and degree sequence D(G) = (d 1 ,d 2 ,... ,d n) , a new measure... more For a graph G of order n , size m and degree sequence D(G) = (d 1 ,d 2 ,... ,d n) , a new measure of irregularity I AG (G) = 1 − n n (d 1 + r)(d 2 + r) ••• (d n + r)/(2m + rn) n , r ∈ R 0 , is introduced. It is shown that if G has maximum I AG-irregularity among all connected graphs of order n and size m , then (i) Δ(G) = n − 1 ; (ii) for each u,v ∈ V (G) with the property d G (u) d G (v) , it holds that N(G,u) ⊆ N[G,v] , where N(G,w) and N[G,w] are the neighbourhood and the closed neighbourhood of w in G , respectively; (iii) G is a threshold graph. Further, it is proven that if a graph H has a minimum value of I AG-irregularity among all irregular graphs of the same order and size, then Δ(H) − δ (H) = 1. Finally, the graphs with minimum and maximum I AG-irregularity in the classes of trees, unicyclic and bicyclic graphs are characterized. Mathematics subject classification (2020): 05C35, 05C07.

arXiv (Cornell University), Jul 4, 2014

The total irregularity of a simple undirected graph G is defined as irr t (G) = 1 2 u,v∈V (G) |d ... more The total irregularity of a simple undirected graph G is defined as irr t (G) = 1 2 u,v∈V (G) |d G (u) − d G (v)|, where d G (u) denotes the degree of a vertex u ∈ V (G). Obviously, irr t (G) = 0 if and only if G is regular. Here, we characterize the non-regular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu, You and Yang [18] about the lower bound on the minimal total irregularity of non-regular connected graphs. We show that the conjectured lower bound of 2n − 4 is attained only if non-regular connected graphs of even order are considered, while the sharp lower bound of n − 1 is attained by graphs of odd order. We also characterize the non-regular graphs with the second and the third smallest total irregularity.

Applied Mathematics and Computation, May 1, 2016

The {\em atom-bond connectivity (ABC) index} is a degree-based graph topological index that found... more The {\em atom-bond connectivity (ABC) index} is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal ABCABCABC index is still an open problem. In~\cite{d-sptmabci-2014}, it was shown that trees with minimal ABC index do not contain so-called {\em BkB_kBk-branches}, with kgeq5k \geq 5kgeq5, and that they do not have more than four B_4B_4B4-branches. Our main results here reveal that the number of B1B_1B1 and B2B_2B2-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not contain more than four B1B_1B1-branches and more than eleven B2B_2B_2-branches.

arXiv (Cornell University), Jun 26, 2017

The atom-bond connectivity (ABC) index is one of the most investigated degreebased molecular stru... more The atom-bond connectivity (ABC) index is one of the most investigated degreebased molecular structure descriptors with a variety of chemical applications. It is known that among all connected graphs, the trees minimize the ABC index. However, a full characterization of trees with a minimal ABC index is still an open problem. By now, one of the proved properties is that a tree with a minimal ABC index may have, at most, one pendent path of length 3, with the conjecture that it cannot be a case if the order of a tree is larger than 1178. Here, we provide an affirmative answer of a strengthened version of that conjecture, showing that a tree with minimal ABC index cannot contain a pendent path of length 3 if its order is larger than 415.

DOAJ (DOAJ: Directory of Open Access Journals), Apr 1, 2019

The variable sum exdeg index of a graph G is defined as SEIa(G) = uv∈E(G) (a d(u) + a d(v)), wher... more The variable sum exdeg index of a graph G is defined as SEIa(G) = uv∈E(G) (a d(u) + a d(v)), where d(u) is the degree of a vertex u and a = 1 is a positive real number. In [1], maximal trees, unicyclic and bicyclic graphs (i.e., graphs with cyclomatic number 0, 1 and 2) and minimal trees and unicyclic graphs (i.e., graphs with cyclomatic number 0 and 1) with respect to variable sum exdeg index for a > 1 were determined. Here, we extend those results in two directions. Firstly, for a > 1, we characterize the extremal graphs with a cyclomatic number k ≤ n − 2, where n is the order of G. Secondly, for 0 < a < 1/e 2 ≈ 0.135335, we characterize the extremal graphs with k ≤ n − 2, and for 0 < a < 1/3, we characterize the trees, unicyclic, bicyclic, tricyclic and tetracyclic graphs having maximal SEIa value.

arXiv (Cornell University), May 6, 2013

The atom-bond connectivity (ABC) index is one of the recently most investigated degree-based mole... more The atom-bond connectivity (ABC) index is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph G, the ABC index is defined as uv∈E(G) (d(u)+d(v)−2) d(u)d(v) , where d(u) is the degree of vertex u in G and E(G) is the set of edges of G. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal ABC index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree sequences of trees and some known properties of trees with minimal ABC index. The obtained results disprove some existing conjectures and suggest new ones to be set.

Applied Mathematics and Computation, Sep 1, 2019

Let G be a simple graph. If all vertices of G have equal degrees, then G is said to be regular. O... more Let G be a simple graph. If all vertices of G have equal degrees, then G is said to be regular. Otherwise, G is irregular. There were various attempts to quantify the irregularity of a graph, of which the Collatz-Sinogowitz index, Bell index, Albertson index, and total irregularity are the best known. We now show that no two of these irregularity measures are mutually consistent, namely that for any two such measures, irr X and irr Y there exist pairs of graphs G 1 , G 2 , such that irr X (G 1) > irr X (G 2) but irr Y (G 1) < irr Y (G 2). This implies that the concept of graph irregularity is not free of ambiguities.

Discrete Applied Mathematics, Apr 1, 2011

For a simple graph G with n vertices and m edges, the inequality M 1 (G)/n ≤ M 2 (G)/m, where M 1... more For a simple graph G with n vertices and m edges, the inequality M 1 (G)/n ≤ M 2 (G)/m, where M 1 (G) and M 2 (G) are the first and the second Zagreb indices of G, is known as Zagreb indices inequality. According to this inequality, a set S of integers is good if for every graph whose degrees of vertices are in S, the inequality holds. We characterize that an interval [a, a + n] is good if and only if a ≥ n(n−1) 2 or [a, a + n] = [1, 4]. We also present an algorithm that decides if an arbitrary set S of cardinality s is good, which requires O(s 2 log s) time and O(s) space.

Discrete Applied Mathematics, May 1, 2016

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemi... more The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal ABC index is still an open problem. In [14], it was shown that trees with minimal ABC index do not contain so-called B k-branches, with k ≥ 5, and that they do not have more than four B 4-branches. Our main results here reveal that the number of B 1 and B 2-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not contain more than four B 1-branches and more than eleven B 2-branches.

Iranian journal of mathematical chemistry, Sep 1, 2018

The reduced reciprocal Randić (RRR) index is a molecular structure descriptor (or more precisely,... more The reduced reciprocal Randić (RRR) index is a molecular structure descriptor (or more precisely, a topological index), which is useful for predicting the standard enthalpy of formation and normal boiling point of isomeric octanes. In this paper, a mathematical aspect of RRR index is explored, or more specifically, the graph(s) having minimum RRR index is/are identified from the collection of all-vertex connected bicyclic graphs for ≥ 5. As a consequence, the best possible lower bound on the RRR index, for-vertex connected bicyclic graphs is obtained when ≥ 5.

arXiv (Cornell University), Jan 23, 2015

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemi... more The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal ABC index is still an open problem. In [14], it was shown that trees with minimal ABC index do not contain so-called B k-branches, with k ≥ 5, and that they do not have more than four B 4-branches. Our main results here reveal that the number of B 1 and B 2-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not contain more than four B 1-branches and more than eleven B 2-branches.

arXiv (Cornell University), Jul 22, 2012

In this note a new measure of irregularity of a simple undirected graph G is introduced. It is na... more In this note a new measure of irregularity of a simple undirected graph G is introduced. It is named the total irregularity of a graph and is defined as irr t (G) = 1 2 u,v∈V (G) |d G (u) − d G (v)| , where d G (u) denotes the degree of a vertex u ∈ V (G). The graphs with maximal total irregularity are determined. It is also shown that among all trees of same order the star graph has the maximal total irregularity.

arXiv (Cornell University), Aug 18, 2016

Measures of the irregularity of chemical graphs could be helpful for QSAR/QSPR studies and for th... more Measures of the irregularity of chemical graphs could be helpful for QSAR/QSPR studies and for the descriptive purposes of biological and chemical properties, such as melting and boiling points, toxicity and resistance. Here we consider the following four established irregularity measures: the irregularity index by Albertson, the total irregularity, the variance of vertex degrees and the Collatz-Sinogowitz index. Through the means of graph structural analysis and derivation, we study the above-mentioned irregularity measures of several chemical molecular graphs which frequently appear in chemical, medical and material engineering, as well as the nanotubes: T U C 4 C 8 (S), T U C 4 C 8 (R), ZigZag T U HC 6 , T U C 4 , Armchair T U V C 6 , then dendrimers T k,d and the circumcoronene series of benzenoid H k. In addition, the irregularities of Mycielski's constructions of cycle and path graphs are analyzed.

Applied Mathematics and Computation

Applied Mathematics and Computation, 2021

Discrete Applied Mathematics, 2018

The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studie... more The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studied vertex-degree-based graph invariants in chemical graph theory. For a given graph G, the ABC index is defined as ∑ uv∈E √ d(u)+d(v)−2 d(u)d(v) , where d(u) is the degree of vertex u in G and E denotes the set of edges of G. In this paper, we present some new structural properties of trees with a minimal ABC index (also refer to as a minimal-ABC tree), which is a step further towards understanding their complete characterization. We show that a minimal-ABC tree cannot simultaneously contain a B 4-branch and B 1 or B 2-branches.

Discrete Applied Mathematics, 2019

All brushes of the extremal graph except for at most one are brooms. Lemma Let B be a non-trivial... more All brushes of the extremal graph except for at most one are brooms. Lemma Let B be a non-trivial brush. If b > a, moving a pendant vertex x from v j to v 1 strictly increases Wex. If b < a, this move strictly decreases Wex, and if b = a, Wex remains the same.

arXiv (Cornell University), 2014

The atom-bond connectivity (ABC) index is a degree-based molecular descriptor, that found chemica... more The atom-bond connectivity (ABC) index is a degree-based molecular descriptor, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal ABC index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results reveal that trees with minimal ABC index do not contain so-called B k-branches, with k ≥ 5, and that they do not have more than four B 4-branches.

Symmetry detection is an important problem with many applications in pattern recognition, compute... more Symmetry detection is an important problem with many applications in pattern recognition, computer vision and computational geometry. In this paper, we propose a novel algorithm for computing a hyperplane of reflexive symmetry of a point set in arbitrary dimension with approximate symmetry. The algorithm is based on the geometric hashing technique. In addition, we consider a relation between the perfect reflective symmetry and the principal components of shapes, a relation that was already a base of few heuristic approaches that tackle the symmetry problem in 2D and 3D. From mechanics, it is known that, if H is a plane of reflective symmetry of the 3D rigid body, then a principal component of the body is orthogonal to H. Here we extend that result to any point set (continuous or discrete) in arbitrary dimension.

Discrete Applied Mathematics, Sep 1, 2023

The problem of characterizing trees with minimal atom-bond-connectivity index (minimal-ABC trees)... more The problem of characterizing trees with minimal atom-bond-connectivity index (minimal-ABC trees) has a reputation as one of the most demanding recent open optimization problems in mathematical chemistry. Here firstly, we give an affirmative answer to the conjecture, which states that enough large minimal-ABC trees are comprised solely of a root vertex and so-called Dz-and Dz+1-branches. Based on the presented theoretical results here and some already known results, we obtain enough constraints to reduce the search space and solve the optimization problem, and thus, determine exactly the minimal-ABC trees of a given arbitrary order.

Mathematical Inequalities & Applications, 2023

For a graph G of order n , size m and degree sequence D(G) = (d 1 ,d 2 ,... ,d n) , a new measure... more For a graph G of order n , size m and degree sequence D(G) = (d 1 ,d 2 ,... ,d n) , a new measure of irregularity I AG (G) = 1 − n n (d 1 + r)(d 2 + r) ••• (d n + r)/(2m + rn) n , r ∈ R 0 , is introduced. It is shown that if G has maximum I AG-irregularity among all connected graphs of order n and size m , then (i) Δ(G) = n − 1 ; (ii) for each u,v ∈ V (G) with the property d G (u) d G (v) , it holds that N(G,u) ⊆ N[G,v] , where N(G,w) and N[G,w] are the neighbourhood and the closed neighbourhood of w in G , respectively; (iii) G is a threshold graph. Further, it is proven that if a graph H has a minimum value of I AG-irregularity among all irregular graphs of the same order and size, then Δ(H) − δ (H) = 1. Finally, the graphs with minimum and maximum I AG-irregularity in the classes of trees, unicyclic and bicyclic graphs are characterized. Mathematics subject classification (2020): 05C35, 05C07.

arXiv (Cornell University), Jul 4, 2014

The total irregularity of a simple undirected graph G is defined as irr t (G) = 1 2 u,v∈V (G) |d ... more The total irregularity of a simple undirected graph G is defined as irr t (G) = 1 2 u,v∈V (G) |d G (u) − d G (v)|, where d G (u) denotes the degree of a vertex u ∈ V (G). Obviously, irr t (G) = 0 if and only if G is regular. Here, we characterize the non-regular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu, You and Yang [18] about the lower bound on the minimal total irregularity of non-regular connected graphs. We show that the conjectured lower bound of 2n − 4 is attained only if non-regular connected graphs of even order are considered, while the sharp lower bound of n − 1 is attained by graphs of odd order. We also characterize the non-regular graphs with the second and the third smallest total irregularity.

Applied Mathematics and Computation, May 1, 2016

The {\em atom-bond connectivity (ABC) index} is a degree-based graph topological index that found... more The {\em atom-bond connectivity (ABC) index} is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal ABCABCABC index is still an open problem. In~\cite{d-sptmabci-2014}, it was shown that trees with minimal ABC index do not contain so-called {\em BkB_kBk-branches}, with kgeq5k \geq 5kgeq5, and that they do not have more than four B_4B_4B4-branches. Our main results here reveal that the number of B1B_1B1 and B2B_2B2-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not contain more than four B1B_1B1-branches and more than eleven B2B_2B_2-branches.

arXiv (Cornell University), Jun 26, 2017

The atom-bond connectivity (ABC) index is one of the most investigated degreebased molecular stru... more The atom-bond connectivity (ABC) index is one of the most investigated degreebased molecular structure descriptors with a variety of chemical applications. It is known that among all connected graphs, the trees minimize the ABC index. However, a full characterization of trees with a minimal ABC index is still an open problem. By now, one of the proved properties is that a tree with a minimal ABC index may have, at most, one pendent path of length 3, with the conjecture that it cannot be a case if the order of a tree is larger than 1178. Here, we provide an affirmative answer of a strengthened version of that conjecture, showing that a tree with minimal ABC index cannot contain a pendent path of length 3 if its order is larger than 415.

DOAJ (DOAJ: Directory of Open Access Journals), Apr 1, 2019

The variable sum exdeg index of a graph G is defined as SEIa(G) = uv∈E(G) (a d(u) + a d(v)), wher... more The variable sum exdeg index of a graph G is defined as SEIa(G) = uv∈E(G) (a d(u) + a d(v)), where d(u) is the degree of a vertex u and a = 1 is a positive real number. In [1], maximal trees, unicyclic and bicyclic graphs (i.e., graphs with cyclomatic number 0, 1 and 2) and minimal trees and unicyclic graphs (i.e., graphs with cyclomatic number 0 and 1) with respect to variable sum exdeg index for a > 1 were determined. Here, we extend those results in two directions. Firstly, for a > 1, we characterize the extremal graphs with a cyclomatic number k ≤ n − 2, where n is the order of G. Secondly, for 0 < a < 1/e 2 ≈ 0.135335, we characterize the extremal graphs with k ≤ n − 2, and for 0 < a < 1/3, we characterize the trees, unicyclic, bicyclic, tricyclic and tetracyclic graphs having maximal SEIa value.

arXiv (Cornell University), May 6, 2013

The atom-bond connectivity (ABC) index is one of the recently most investigated degree-based mole... more The atom-bond connectivity (ABC) index is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph G, the ABC index is defined as uv∈E(G) (d(u)+d(v)−2) d(u)d(v) , where d(u) is the degree of vertex u in G and E(G) is the set of edges of G. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal ABC index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree sequences of trees and some known properties of trees with minimal ABC index. The obtained results disprove some existing conjectures and suggest new ones to be set.

Applied Mathematics and Computation, Sep 1, 2019

Let G be a simple graph. If all vertices of G have equal degrees, then G is said to be regular. O... more Let G be a simple graph. If all vertices of G have equal degrees, then G is said to be regular. Otherwise, G is irregular. There were various attempts to quantify the irregularity of a graph, of which the Collatz-Sinogowitz index, Bell index, Albertson index, and total irregularity are the best known. We now show that no two of these irregularity measures are mutually consistent, namely that for any two such measures, irr X and irr Y there exist pairs of graphs G 1 , G 2 , such that irr X (G 1) > irr X (G 2) but irr Y (G 1) < irr Y (G 2). This implies that the concept of graph irregularity is not free of ambiguities.

Discrete Applied Mathematics, Apr 1, 2011

For a simple graph G with n vertices and m edges, the inequality M 1 (G)/n ≤ M 2 (G)/m, where M 1... more For a simple graph G with n vertices and m edges, the inequality M 1 (G)/n ≤ M 2 (G)/m, where M 1 (G) and M 2 (G) are the first and the second Zagreb indices of G, is known as Zagreb indices inequality. According to this inequality, a set S of integers is good if for every graph whose degrees of vertices are in S, the inequality holds. We characterize that an interval [a, a + n] is good if and only if a ≥ n(n−1) 2 or [a, a + n] = [1, 4]. We also present an algorithm that decides if an arbitrary set S of cardinality s is good, which requires O(s 2 log s) time and O(s) space.

Discrete Applied Mathematics, May 1, 2016

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemi... more The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal ABC index is still an open problem. In [14], it was shown that trees with minimal ABC index do not contain so-called B k-branches, with k ≥ 5, and that they do not have more than four B 4-branches. Our main results here reveal that the number of B 1 and B 2-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not contain more than four B 1-branches and more than eleven B 2-branches.

Iranian journal of mathematical chemistry, Sep 1, 2018

The reduced reciprocal Randić (RRR) index is a molecular structure descriptor (or more precisely,... more The reduced reciprocal Randić (RRR) index is a molecular structure descriptor (or more precisely, a topological index), which is useful for predicting the standard enthalpy of formation and normal boiling point of isomeric octanes. In this paper, a mathematical aspect of RRR index is explored, or more specifically, the graph(s) having minimum RRR index is/are identified from the collection of all-vertex connected bicyclic graphs for ≥ 5. As a consequence, the best possible lower bound on the RRR index, for-vertex connected bicyclic graphs is obtained when ≥ 5.

arXiv (Cornell University), Jan 23, 2015

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemi... more The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal ABC index is still an open problem. In [14], it was shown that trees with minimal ABC index do not contain so-called B k-branches, with k ≥ 5, and that they do not have more than four B 4-branches. Our main results here reveal that the number of B 1 and B 2-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not contain more than four B 1-branches and more than eleven B 2-branches.

arXiv (Cornell University), Jul 22, 2012

In this note a new measure of irregularity of a simple undirected graph G is introduced. It is na... more In this note a new measure of irregularity of a simple undirected graph G is introduced. It is named the total irregularity of a graph and is defined as irr t (G) = 1 2 u,v∈V (G) |d G (u) − d G (v)| , where d G (u) denotes the degree of a vertex u ∈ V (G). The graphs with maximal total irregularity are determined. It is also shown that among all trees of same order the star graph has the maximal total irregularity.

arXiv (Cornell University), Aug 18, 2016

Measures of the irregularity of chemical graphs could be helpful for QSAR/QSPR studies and for th... more Measures of the irregularity of chemical graphs could be helpful for QSAR/QSPR studies and for the descriptive purposes of biological and chemical properties, such as melting and boiling points, toxicity and resistance. Here we consider the following four established irregularity measures: the irregularity index by Albertson, the total irregularity, the variance of vertex degrees and the Collatz-Sinogowitz index. Through the means of graph structural analysis and derivation, we study the above-mentioned irregularity measures of several chemical molecular graphs which frequently appear in chemical, medical and material engineering, as well as the nanotubes: T U C 4 C 8 (S), T U C 4 C 8 (R), ZigZag T U HC 6 , T U C 4 , Armchair T U V C 6 , then dendrimers T k,d and the circumcoronene series of benzenoid H k. In addition, the irregularities of Mycielski's constructions of cycle and path graphs are analyzed.

Applied Mathematics and Computation

Applied Mathematics and Computation, 2021

Discrete Applied Mathematics, 2018

The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studie... more The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studied vertex-degree-based graph invariants in chemical graph theory. For a given graph G, the ABC index is defined as ∑ uv∈E √ d(u)+d(v)−2 d(u)d(v) , where d(u) is the degree of vertex u in G and E denotes the set of edges of G. In this paper, we present some new structural properties of trees with a minimal ABC index (also refer to as a minimal-ABC tree), which is a step further towards understanding their complete characterization. We show that a minimal-ABC tree cannot simultaneously contain a B 4-branch and B 1 or B 2-branches.

Discrete Applied Mathematics, 2019

All brushes of the extremal graph except for at most one are brooms. Lemma Let B be a non-trivial... more All brushes of the extremal graph except for at most one are brooms. Lemma Let B be a non-trivial brush. If b > a, moving a pendant vertex x from v j to v 1 strictly increases Wex. If b < a, this move strictly decreases Wex, and if b = a, Wex remains the same.

arXiv (Cornell University), 2014

The atom-bond connectivity (ABC) index is a degree-based molecular descriptor, that found chemica... more The atom-bond connectivity (ABC) index is a degree-based molecular descriptor, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal ABC index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results reveal that trees with minimal ABC index do not contain so-called B k-branches, with k ≥ 5, and that they do not have more than four B 4-branches.