International Scholarly Research Notices, Nov 3, 2014
There is an age-old question in all branches of network analysis. What makes an actor in a networ... more There is an age-old question in all branches of network analysis. What makes an actor in a network important, courted, or sought? Both Crossley and Bonacich contend that rather than its intrinsic wealth or value, an actor's status lies in the structures of its interactions with other actors. Since pairwise relation data in a network can be stored in a two-dimensional array or matrix, graph theory and linear algebra lend themselves as great tools to gauge the centrality (interpreted as importance, power, or popularity, depending on the purpose of the network) of each actor. We express known and new centralities in terms of only two matrices associated with the network. We show that derivations of these expressions can be handled exclusively through the main eigenvectors (not orthogonal to the all-one vector) associated with the adjacency matrix. We also propose a centrality vector (SWIPD) which is a linear combination of the square, walk, power, and degree centrality vectors with weightings of the various centralities depending on the purpose of the network. By comparing actors' scores for various weightings, a clear understanding of which actors are most central is obtained. Moreover, for threshold networks, the (SWIPD) measure turns out to be independent of the weightings.
Springer proceedings in mathematics & statistics, 2020
Two methods, structural (constructive) and multiplier (analytical), of exact enumeration of undir... more Two methods, structural (constructive) and multiplier (analytical), of exact enumeration of undirected and directed circulant graphs of orders 27 and 125 are elaborated and represented in detail here together with intermediate and final numerical data and generating functions. The first method is based on the classification of circulant graphs in terms of S-rings and results in exhaustive listing (with the aid of COCO and GAP) of all corresponding S-rings of the indicated orders. The latter method is based on a general theoretical approach developed earlier for counting circulant graphs of prime-power orders. It is a Redfield–Polya type of enumeration based on an isomorphism criterion for circulant graphs of such orders. In particular, five intermediate enumeration subproblems arise, which are refined further into eleven subproblems of this type. We give a brief survey of some background theory of the results which form the basis of our computational approach. Some curious and rather unexpected identities are established between intermediate valency-specified enumerators and their validity is conjectured for arbitrary cubed odd primes \(p^3\). We conclude with the enumeration of self-complementary circulant graphs of orders 27 and 125. We believe that this research can serve as the crucial step towards the explicit uniform enumeration formulae for circulant graphs of orders \(p^3\) for arbitrary primes \(p>2\).
The reconstruction number rn(G) of a graph G was introduced by Harary and Plantholt as the smalle... more The reconstruction number rn(G) of a graph G was introduced by Harary and Plantholt as the smallest number of vertex-deleted subgraphs G i = G-vi in the deck of G which do not all appear in the deck of any other graph. For any graph theoretic property ~, Harary defined the ~reconstruction number of a graph G ~ ~ as the smallest number of the G i in the deck of G, which do not all appear in the deck of any other graph in ~. We now study the maximal planar graph reconstruction number J/lrn(G), proving that its value is either 1 or 2 and characterizing those with value 1.
One popular stopping rule intended to increase the rate of male births is to keep having babies u... more One popular stopping rule intended to increase the rate of male births is to keep having babies until the first male is born. We show that such a stopping rule does not change the sex ratio at birth when the probability p that a birth produces a male is constant across a population. We show, however, that when p varies across couples, as evidence suggests that it does in the human population, with mean around 0.515 and standard deviation 0.05, then such a stopping rule tends to favour female births and the correct rule should be to stop procreating until the first female is born. But we also show that it does not seem that such stopping rules, even if favouring male births, can account for the dramatic increase in the rate of male births registered in some countries. Most of these conclusions have appeared in some form or other in various studies, but they are scattered across the literature and are very often presented using some heavy mathematical techniques. In this paper, we try to bring together these ideas and make them more accessible by analysing these stopping rules using the simplest mathematical tools possible.
In this note we give an elementary combinatorial proof of a formula of Macris and Pul6 for the nu... more In this note we give an elementary combinatorial proof of a formula of Macris and Pul6 for the number of Euler trails in a digraph all of whose vertices have in-degree and out-degree equal to2.
This in-depth coverage of important areas of graph theory maintains a focus on symmetry propertie... more This in-depth coverage of important areas of graph theory maintains a focus on symmetry properties of graphs. Standard topics on graph automorphisms are presented early on, while in later chapters more specialised topics are tackled, such as graphical regular representations and pseudosimilarity. The final four chapters are devoted to the reconstruction problem, and here special emphasis is given to those results that involve the symmetry of graphs, many of which are not to be found in other books. This second edition expands on several of the topics found in the first edition and includes both an enriched bibliography and a wide collection of exercises. Clearer proofs are provided, as are new examples of graphs with interesting symmetry properties. Any student who masters the contents of this book will be well prepared for current research in many aspects of the theory of graph automorphisms and the reconstruction problem.
This paper provides a model of the use of computer algebra experimentation in algebraic graph the... more This paper provides a model of the use of computer algebra experimentation in algebraic graph theory. Starting from the semisymmetric cubic graph L on 112 vertices, we embed it into another semisymmetric graph N of valency 15 on the same vertex set. In order to consider systematically the links between L and N a number of combinatorial structures are involved and related coherent configurations are investigated. In particular, the construction of the incidence double cover of directed graphs is exploited. As a natural by-product of the approach presented here, a number of new interesting (mostly non-Schurian) association schemes on 56, 112 and 120 vertices are introduced and briefly discussed. We use computer algebra system GAP (including GRAPE and nauty), as well as computer package COCO.
Journal of Combinatorial Theory, Series B, Apr 1, 1981
In the first paper 131, the author, together with Fiorini. has shown that maximal planar graphs a... more In the first paper 131, the author, together with Fiorini. has shown that maximal planar graphs are recognizable from their decks of vertex-deleted subgraphs. The aim of this paper is to show that such graphs are reconstructible. We first define an ordinary vertex to be a vertex whose valency is at least 4. Now, given a maximal planar graph G of minimum valency 3. we can recognize the maximal planarity of G from the deck g(G) 13). Therefore for any ordinary vertex v, we need only consider the p(v)-representations of G
International Scholarly Research Notices, Nov 3, 2014
There is an age-old question in all branches of network analysis. What makes an actor in a networ... more There is an age-old question in all branches of network analysis. What makes an actor in a network important, courted, or sought? Both Crossley and Bonacich contend that rather than its intrinsic wealth or value, an actor's status lies in the structures of its interactions with other actors. Since pairwise relation data in a network can be stored in a two-dimensional array or matrix, graph theory and linear algebra lend themselves as great tools to gauge the centrality (interpreted as importance, power, or popularity, depending on the purpose of the network) of each actor. We express known and new centralities in terms of only two matrices associated with the network. We show that derivations of these expressions can be handled exclusively through the main eigenvectors (not orthogonal to the all-one vector) associated with the adjacency matrix. We also propose a centrality vector (SWIPD) which is a linear combination of the square, walk, power, and degree centrality vectors with weightings of the various centralities depending on the purpose of the network. By comparing actors' scores for various weightings, a clear understanding of which actors are most central is obtained. Moreover, for threshold networks, the (SWIPD) measure turns out to be independent of the weightings.
Springer proceedings in mathematics & statistics, 2020
Two methods, structural (constructive) and multiplier (analytical), of exact enumeration of undir... more Two methods, structural (constructive) and multiplier (analytical), of exact enumeration of undirected and directed circulant graphs of orders 27 and 125 are elaborated and represented in detail here together with intermediate and final numerical data and generating functions. The first method is based on the classification of circulant graphs in terms of S-rings and results in exhaustive listing (with the aid of COCO and GAP) of all corresponding S-rings of the indicated orders. The latter method is based on a general theoretical approach developed earlier for counting circulant graphs of prime-power orders. It is a Redfield–Polya type of enumeration based on an isomorphism criterion for circulant graphs of such orders. In particular, five intermediate enumeration subproblems arise, which are refined further into eleven subproblems of this type. We give a brief survey of some background theory of the results which form the basis of our computational approach. Some curious and rather unexpected identities are established between intermediate valency-specified enumerators and their validity is conjectured for arbitrary cubed odd primes \(p^3\). We conclude with the enumeration of self-complementary circulant graphs of orders 27 and 125. We believe that this research can serve as the crucial step towards the explicit uniform enumeration formulae for circulant graphs of orders \(p^3\) for arbitrary primes \(p>2\).
The reconstruction number rn(G) of a graph G was introduced by Harary and Plantholt as the smalle... more The reconstruction number rn(G) of a graph G was introduced by Harary and Plantholt as the smallest number of vertex-deleted subgraphs G i = G-vi in the deck of G which do not all appear in the deck of any other graph. For any graph theoretic property ~, Harary defined the ~reconstruction number of a graph G ~ ~ as the smallest number of the G i in the deck of G, which do not all appear in the deck of any other graph in ~. We now study the maximal planar graph reconstruction number J/lrn(G), proving that its value is either 1 or 2 and characterizing those with value 1.
One popular stopping rule intended to increase the rate of male births is to keep having babies u... more One popular stopping rule intended to increase the rate of male births is to keep having babies until the first male is born. We show that such a stopping rule does not change the sex ratio at birth when the probability p that a birth produces a male is constant across a population. We show, however, that when p varies across couples, as evidence suggests that it does in the human population, with mean around 0.515 and standard deviation 0.05, then such a stopping rule tends to favour female births and the correct rule should be to stop procreating until the first female is born. But we also show that it does not seem that such stopping rules, even if favouring male births, can account for the dramatic increase in the rate of male births registered in some countries. Most of these conclusions have appeared in some form or other in various studies, but they are scattered across the literature and are very often presented using some heavy mathematical techniques. In this paper, we try to bring together these ideas and make them more accessible by analysing these stopping rules using the simplest mathematical tools possible.
In this note we give an elementary combinatorial proof of a formula of Macris and Pul6 for the nu... more In this note we give an elementary combinatorial proof of a formula of Macris and Pul6 for the number of Euler trails in a digraph all of whose vertices have in-degree and out-degree equal to2.
This in-depth coverage of important areas of graph theory maintains a focus on symmetry propertie... more This in-depth coverage of important areas of graph theory maintains a focus on symmetry properties of graphs. Standard topics on graph automorphisms are presented early on, while in later chapters more specialised topics are tackled, such as graphical regular representations and pseudosimilarity. The final four chapters are devoted to the reconstruction problem, and here special emphasis is given to those results that involve the symmetry of graphs, many of which are not to be found in other books. This second edition expands on several of the topics found in the first edition and includes both an enriched bibliography and a wide collection of exercises. Clearer proofs are provided, as are new examples of graphs with interesting symmetry properties. Any student who masters the contents of this book will be well prepared for current research in many aspects of the theory of graph automorphisms and the reconstruction problem.
This paper provides a model of the use of computer algebra experimentation in algebraic graph the... more This paper provides a model of the use of computer algebra experimentation in algebraic graph theory. Starting from the semisymmetric cubic graph L on 112 vertices, we embed it into another semisymmetric graph N of valency 15 on the same vertex set. In order to consider systematically the links between L and N a number of combinatorial structures are involved and related coherent configurations are investigated. In particular, the construction of the incidence double cover of directed graphs is exploited. As a natural by-product of the approach presented here, a number of new interesting (mostly non-Schurian) association schemes on 56, 112 and 120 vertices are introduced and briefly discussed. We use computer algebra system GAP (including GRAPE and nauty), as well as computer package COCO.
Journal of Combinatorial Theory, Series B, Apr 1, 1981
In the first paper 131, the author, together with Fiorini. has shown that maximal planar graphs a... more In the first paper 131, the author, together with Fiorini. has shown that maximal planar graphs are recognizable from their decks of vertex-deleted subgraphs. The aim of this paper is to show that such graphs are reconstructible. We first define an ordinary vertex to be a vertex whose valency is at least 4. Now, given a maximal planar graph G of minimum valency 3. we can recognize the maximal planarity of G from the deck g(G) 13). Therefore for any ordinary vertex v, we need only consider the p(v)-representations of G
Social Psychology: New Research (Lamont, Ed.), 2009
Social representations is a relatively new area of research which is rapidly becoming an importan... more Social representations is a relatively new area of research which is rapidly becoming an important tool in understanding social behavior. In this chapter we will be using this theory to understand how university students in Malta look upon religion. A self-administered questionnaire was given to a random sample of 650 students at the University of Malta, of which 421 completed the questionnaire. The questionnaire consisted of 35 questions about religious attitudes and behavior, and also included the Post-Critical Belief Scale (Duriez et al., 2005). The data were subjected to statistical analysis. The results were used to identify the different social representations which students had of religion. The implications for understanding how religious beliefs correlate with religious behavior among students in tertiary education will be discussed.
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