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A Survey of Research on Automated Mathematical ConjectureMaking
 FAJTLOWICZ (EDITORS), AMERICAN MATHEMATICAL SOCIETY
, 2005
"... The first attempt at automating mathematical conjecturemaking appeared in the late1950s. It was not until the mid1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlyi ..."
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The first attempt at automating mathematical conjecturemaking appeared in the late1950s. It was not until the mid1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlying this program is the Principle of the Strongest Conjecture: make the strongest conjecture for which no counterexample is known. These two programs as well as other attempts to automate mathematical conjecturemaking are surveyed—the success of a conjecturemaking program, it is found, correlates strongly whether the program is designed to produce statements that are relevant to answering or advancing our mathematical questions.
Elphick: Three New/Old VertexDegreeBased Topological Indices
 MATCH Commun. Math. Comput. Chem
"... Abstract Three vertexdegreebased graph invariants are presented, that earlier have been considered in the chemical and/or mathematical literature, but that evaded the attention of most mathematical chemists. These are the reciprocal Randić index (RR), the reduced second Zagreb index RM 2 , and th ..."
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Abstract Three vertexdegreebased graph invariants are presented, that earlier have been considered in the chemical and/or mathematical literature, but that evaded the attention of most mathematical chemists. These are the reciprocal Randić index (RR), the reduced second Zagreb index RM 2 , and the reduced reciprocal Randić index (RRR) We outline the literature sources of these topological indices, their main mathematical properties, and establish their correlating abilities w.r.t. characteristic physicochemical properties of alkanes.
On the Randic ́ index of cacti
"... The Randic ́ index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))− 1 2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. In the paper, we give a sharp lower bound on the Randić index of cacti. ..."
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The Randic ́ index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))− 1 2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. In the paper, we give a sharp lower bound on the Randić index of cacti.
The Total Irregularity of Graphs under Graph Operations
, 2014
"... The total irregularity of a graph G is defined as irrt(G) = ..."
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The total irregularity of a graph G is defined as irrt(G) =
Digenes: genetic algorithms to discover conjectures about directed and undirected graphs
"... Abstract. We present Digenes, a new discovery system that aims to help researchers in graph theory. While its main task is to find extremal graphs for a given (function of) invariants, it also provides some basic support in proof conception. This has already been proved to be very useful to find new ..."
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Abstract. We present Digenes, a new discovery system that aims to help researchers in graph theory. While its main task is to find extremal graphs for a given (function of) invariants, it also provides some basic support in proof conception. This has already been proved to be very useful to find new conjectures since the AutoGraphiX system of Caporossi and Hansen [8]. However, unlike existing systems, Digenes can be used both with directed or undirected graphs. In this paper, we present the principles and functionality of Digenes, describe the genetic algorithms that have been designed to achieve them, and give some computational results and open questions. This do arise some interesting questions regarding genetic algorithms design particular to this field, such as crossover definition.
Games of Dynamic Network Formation
"... We combine a network game introduced in Ballester et al. (2006), where the Nash equilibrium action of each agent is proportional to her Bonacich centrality (Bonacich, 1987), with an endogenous network formation process. Links are formed on the basis of centrality while the network is exposed to a v ..."
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We combine a network game introduced in Ballester et al. (2006), where the Nash equilibrium action of each agent is proportional to her Bonacich centrality (Bonacich, 1987), with an endogenous network formation process. Links are formed on the basis of centrality while the network is exposed to a volatile environment introducing interruptions in the connections between agents. Taking into account bounded rational decision making, new links are formed to the neighbors’ neighbors with the highest centrality. The volatile environment causes existing links to decay to the neighbor with the lowest centrality. We show analytically that there exist stationary networks and that their topological properties completely match with features exhibited by social and economic networks. Moreover, we find that there exists a sharp transition in efficiency and network density from highly centralized to decentralized networks.
ON SOME INTERCONNECTIONS BETWEEN COMBINATORIAL OPTIMIZATION AND EXTREMAL GRAPH THEORY
, 2004
"... The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified type ..."
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The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the
Comparing the Zagreb Indices*
, 2006
"... Let G = (V, E) be a simple graph with n = V  vertices and m = E  edges; let d1, d2, …, dn denote the degrees of the vertices of G. If D = max i id ≤ 4, G is a chemical graph. The first and second Zagreb indices are defined as M1 = di i ..."
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Let G = (V, E) be a simple graph with n = V  vertices and m = E  edges; let d1, d2, …, dn denote the degrees of the vertices of G. If D = max i id ≤ 4, G is a chemical graph. The first and second Zagreb indices are defined as M1 = di i