On the Zagreb Indices Equality

For a simple graph $G$ with $n$ vertices and $m$ edges, the first Zagreb index and the second Zagreb index are defined as $M_1(G)=\sum_{v\in V}d(v)^2$ and $M_2(G)=\sum_{uv\in E}d(u)d(v)$. In \cite{VGFAD}, it was shown that if a connected graph $G$ has maximal degree 4, then $G$ satisfies $M_1(G)/n = M_2(G)/m$ (also known as the Zagreb indices equality) if and only if $G$ is regular or biregular of class 1 (a biregular graph whose no two vertices of same degree are adjacent). There, it was also shown that there exist infinitely many connected graphs of maximal degree $\Delta= 5$ that are neither regular nor biregular of class 1 which satisfy the Zagreb indices equality. Here, we generalize that result by showing that there exist infinitely many connected graphs of maximal degree $\Delta \geq 5$ that are neither regular nor biregular graphs of class 1 which satisfy the Zagreb indices equality. We also consider when the above equality holds when the degrees of vertices of a given graph are in a prescribed interval of integers.Comment: 11 pages, 1 figur

Algorithm and Complexity for a Network Assortativity Measure

We show that finding a graph realization with the minimum Randi\'c index for a given degree sequence is solvable in polynomial time by formulating the problem as a minimum weight perfect b-matching problem. However, the realization found via this reduction is not guaranteed to be connected. Approximating the minimum weight b-matching problem subject to a connectivity constraint is shown to be NP-Hard. For instances in which the optimal solution to the minimum Randi\'c index problem is not connected, we describe a heuristic to connect the graph using pairwise edge exchanges that preserves the degree sequence. In our computational experiments, the heuristic performs well and the Randi\'c index of the realization after our heuristic is within 3% of the unconstrained optimal value on average. Although we focus on minimizing the Randi\'c index, our results extend to maximizing the Randi\'c index as well. Applications of the Randi\'c index to synchronization of neuronal networks controlling respiration in mammals and to normalizing cortical thickness networks in diagnosing individuals with dementia are provided.Comment: Added additional section on application

Scale-free spanning trees: complexity, bounds and algorithms

We introduce and study the general problem of finding a most "scale-free-like" spanning tree of a connected graph. It is motivated by a particular problem in epidemiology, and may be useful in studies of various dynamical processes in networks. We employ two possible objective functions for this problem and introduce the corresponding algorithmic problems termed $m$-SF and $s$-SF Spanning Tree problems. We prove that those problems are APX- and NP-hard, respectively, even in the classes of cubic, bipartite and split graphs. We study the relations between scale-free spanning tree problems and the max-leaf spanning tree problem, which is the classical algorithmic problem closest to ours. For split graphs, we explicitly describe the structure of optimal spanning trees and graphs with extremal solutions. Finally, we propose two Integer Linear Programming formulations and two fast heuristics for the $s$-SF Spanning Tree problem, and experimentally assess their performance using simulated and real data

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

Essays on the Network Analysis of Culture

Nelle relazioni economiche, negli accordi internazionali e nel dialogo istituzionale, la parola distanza \ue8 una delle pi\uf9 enunciate. Ci sono distanze esogene da colmare per creare legami, a volte ci sono chiusure necessarie e altre volte rotture inevitabili, ma questo pu\uf2 dipendere, cos\uec come le distanze geografiche e fisiche, e gli interessi impliciti, in gran parte dallo status culturale di gruppi di individui. La valutazione quantitativa della distanza tra due entit\ue0 \ue8 una propriet\ue0 diadica ed in quanto tale, la presenza, intensit\ue0, direzione e segno di un legame rappresenta un modo per catturarla. Poich\ue9 le entit\ue0 possono essere individui, oggetti, societ\ue0, paesi, pianeti, cos\uec come reti che si riferiscono a contesti specifici, e il modo di misurare la somiglianza tra di loro pu\uf2 essere vario, una cosa peculiare delle distanze \ue8 la loro natura mutevole. Mentre le distanze fisiche sono quasi oggettivamente calcolabili, nel caso della cultura (ed anche di altri concetti pi\uf9 o meno ampi) l\u2019utilizzo di un metodo rispetto ad un altro potrebbe cambiare radicalmente la relazione di distanza tra le entit\ue0, soprattutto se esse hanno un alto grado di complessit\ue0. Il bagaglio culturale svolge un ruolo importante nel determinare lo status socio-economico di un paese e la sua caratterizzazione in termini di somiglianza con altri paesi. Il Capitolo 1 - utilizzando i dati della WVS/EVS Joint 2017 - operativizza una definizione di cultura che tiene conto delle interdipendenze tra tratti culturali a livello di paese e propone una nuova misura di distanza culturale. Sfruttando un recente algoritmo Bayesiano di Copula Gaussian graphical models, questo Capitolo stima per ciascuno di 76 paesi inclusi nella WVS/EVS Joint 2017, la rete culturale di interdipendenze tra tratti culturali considerando diversi insiemi di essi: i 6 della prima batteria di domande, i 10 della mappa culturale di Inglehart-Welzel, i 14 della mappa culturale di Inglehart-Welzel, dove per gli indici di \u201cPost-materialism\u201d e \u201cAutonomy\u201d sono state utilizzate le variabili da cui sono ricavate, e 60 tratti culturali dei quali, 14 come definiti in precedenza, 6 fanno riferimento alla prima batteria di domande e i restanti 40 sono selezionati in modo da ottenere un numero di variabili che possa far fronte al trade-off tra il tempo di elaborazione dell\u2019algoritmo e il minimo numero di valori mancanti per paese. Dopo aver definito le distanze tra i paesi considerando sia le reti culturali che le distribuzioni dei tratti culturali, attraverso il metodo DISTATIS, questo Capitolo osserva come l'aggiunta della componente di rete a quella distributiva classica, modifichi sostanzialmente la misura della distanza culturale sia nel caso di pochi tratti culturali (6, 10 e 14) che nel caso di pi\uf9 tratti culturali (60). Infine, esso afferma che la struttura di rete della cultura nazionale \ue8 importante per la definizione della distanza culturale tra i paesi del mondo e trova due misure finali di distanza: il Compromise_Large (da 60 variabili) e il Compromise_IW (dalle variabili della mappa culturale di Inglehart-Welzel). L'effetto delle variabili culturali sulla situazione economica di un paese, o pi\uf9 in generale di un'area geograficamente definita, \ue8 stato negli ultimi anni scandagliato dalla letteratura economica. Le distanze culturali, genetiche, geografiche, climatiche, semantiche, etniche, linguistiche, politiche sono state spesso incluse nei modelli econometrici come variabili indipendenti o di controllo. Il Capitolo 2 segue questa letteratura, prima confrontando individualmente tre misurazioni della distanza culturale calcolate nel Capitolo 1 con altre distanze usate in letteratura assieme alla distanza culturale o come proxy di essa, e poi confrontandole (le misure di distanza culturale e quelle dalla letteratura) congiuntamente tramite DISTATIS. Le tre distanze culturali sono le due nuove misure di cui sopra (Compromise_Large e Compromise_IW) e l'IW index ottenuto come distanza euclidea tra i paesi nella mappa culturale di Inglehart-Welzel, mentre le altre distanze prendono in considerazione la condizione climatica, l'etnia e la lingua, la genetica ed il recente fenomeno di Facebook. Infine, questo Capitolo considera tutte le misure di distanza all\u2019interno di un Social Relations Regression Model (SRRM) che stima la distanza tra i paesi in base al PIL pro capite (anno 2017). Il risultato finale mostra che le distanze culturali sono poco correlate con le distanze prese dalla letteratura, e quando si trova un compromesso tra di loro, di solito la Compromise_Large \ue8 caratterizzata da un peso leggermente superiore. La conclusione principale riguarda l'importante potere esplicativo della distanza Compromise_Large sulla distanza in PIL pro capite rispetto a quello della IW index e della Compromise_IW, la quale ha un significato intermedio tra le due. Ci\uf2 conferma l'importanza di considerare la rete culturale nazionale di interdipendenze tra tratti culturali nella definizione generale della distanza culturale, ed anche che l\u2019aggiunta di un numero maggiore di tratti culturali pu\uf2 influire nella sua specificazione, seppur i tratti culturali considerati da Ronald Inglehart e Christian Welzel nella costruzione della loro mappa culturale sembrano catturare gi\ue0 una buona parte dell\u2019informazione culturale dei paesi. La produzione abnorme di dati nel nostro tempo ha permesso l'osservazione di grandi collezioni di reti all\u2019interno di un campo di analisi specifico, le quali possono essere caratterizzate anche da una diversa dimensione l\u2019una dall\u2019altra (ad esempio si pu\uf2 pensare alla rete commerciale tra paesi di ogni prodotto). Una rete \ue8 un oggetto complesso, per cui un modo comune per analizzare e comparare congiuntamente un set di reti \ue8 ridurne la complessit\ue0 proiettandole in uno spazio ridotto attraverso i descrittori che le caratterizzano. \uc8 qui che sorge il problema analizzato nel Capitolo 3: qual \ue8 il sottoinsieme di descrittori che mantiene le caratteristiche delle reti il pi\uf9 possibile invariate nel processo di mapping, ovvero proietta in punti diversi dello spazio reti non isomorfe e raggruppa vicine reti strutturalmente simili tra di loro e lontano reti dissimili? Attraverso una simulazione di reti da quattro modelli generativi (Random, Scale-free, Small-world e Stochastic block model) e la selezione di un ampio insieme di descrittori riferenti ai livelli micro, meso e macro di analisi della rete, questo Capitolo trova tramite il metodo di Subgroup Discovery un piccolo sottoinsieme di descrittori. Questo sottoinsieme \ue8 composto da 5 descrittori: il momento primo del Coefficiente di Clustering Locale, 3 configurazioni di Motifs e il descrittore di Smallworldness. L'efficacia dei descrittori \ue8 valutata applicandoli all'insieme delle reti culturali binarie con 60 tratti culturali stimate nel Capitolo 1 e confrontando le distanze tra questi punti-rete nello spazio dei descrittori con distanze di reti popolari in letteratura. Le principali innovazioni sono due: la costruzione di un nuovo indice di distanza culturale tra i paesi, in cui \ue8 inclusa la rete culturale di interdipendenze tra tratti culturali; la selezione di un piccolo sottoinsieme efficiente di descrittori per la proiezione nello spazio di insiemi di reti binarie che possono avere grandezza diversa l\u2019una dall\u2019altra.In economic relations, in international agreements and in institutional dialogue, the word distance is one of the most enunciated. There are exogenous distances to be bridged to ignite a bond, sometimes there are necessary cracks and other times unavoidable breaks, but this may depend, as well as geographical and physical distances, and implicit interests, largely on the cultural status of groups of individuals. The quantitative evaluation of the distance between two entities is a dyadic property and as such, the presence, intensity, direction and sign of their tie is a way to undertake it. Since entities can be individuals, objects, companies, countries, planets, as well as networks referring to specific contexts, and the way to measure similarity between them is various, a peculiarity thing of distances is their changeable nature. While physical distances are almost objectively computable, in case of culture (and even other more or less broad concepts) using a method rather than another could radically change the proximity relationship between entities, especially if they have a high degree of complexity. The cultural background plays an important role in determining the socio-economic status of a country and its characterization in terms of similarity to other countries. The Chapter 1 - using data from the WVS/EVS Joint 2017 - operationalizes a definition of culture that takes into account the interdependencies between cultural traits at country level and calculates a new measure of cultural distance. Taking advantage of a recent Bayesian algorithm by Gaussian copula graphical model, this Chapter estimates for each of 76 countries included in the WVS/EVS Joint 2017, the cultural network of interdependencies between cultural traits considering different sets of them: the 6 from the first battery of questions, the 10 of the Inglehart-Welzel Cultural Map, the 14 of the Inglehart-Welzel Cultural Map, where for \u201cPost-materialism\u201d and \u201cAutonomy\u201d indices are used the variables from which they are derived, and 60 cultural traits of which, 14 as previously defined, 6 refer to the first battery of questions and the remaining 40 are selected to get a number that can cope with the trade-off between processing time and the minimum number of missing values per country. After defining the distances between countries considering both cultural networks and distributions of cultural traits, this Chapter observes via DISTATIS how the addition of the network component to the classic distributional one, substantially modifies the measure of cultural distance both in the case of a few cultural traits (6, 10 and 14) and in the case of more cultural traits (60). Finally, it affirms that the network structure of the national culture matters for the definition of the cultural distance among worldwide countries and finds two final distance measures: Compromise_Large (from 60 variables) and Compromise_IW (from the Inglehart-Welzel cultural map variables). The effect of cultural variables on the economic situation of a country or more generally of a geographically definable area, has been scoured in recent years by the economic literature. Cultural, genetic, geographical, climatic, semantic, ethnic, linguistic, political distances have often been included in econometric models as independent or control variables. The Chapter 2 follows this literature, firstly by individually comparing three measurements of cultural distance calculated in Chapter 1 with other distances used in literature together with cultural distance or as a proxy of it, and secondly by jointly comparing them (the measurements of cultural distance and those from literature) via DISTATIS. The three cultural distances are the two new measures mentioned above (Compromise_Large and Compromise_IW) and the IW index obtained as Euclidean distance between countries in the Inglehart-Welzel cultural map, while the other distances take into consideration climatic condition, ethnicity and language, genetics and the recent phenomenon of Facebook. Finally, this Chapter considers these distance measures into a Social Relations Regression Model (SRRM) which estimates the distance between countries in GDP per capita (year 2017). The final result shows that cultural distances are poorly correlated with the distances from the literature, and when a compromise is found between them, usually the Compromise_Large is characterized by a slightly higher weight. The main conclusion concerns the important explanatory power of the Compromise_Large distance on the distance in GDP per capita compared to that of the IW index and the Compromise_IW, which has an intermediate meaning between the two. This confirms the importance of considering the national cultural network of interdependencies between cultural traits in the overall definition of cultural distance, and also that the addition of more cultural traits may influence its specification, although the cultural traits considered by Inglehart and Welzel in the construction of their cultural map seem to capture already a good part of the cultural information of the countries. The abnormal production of data in our time has allowed the observation of large collections of networks within a specific field of analysis, which can also be characterized by a different size from each other, e.g. you can think of the trade network of each product between countries. A network is a complex object, so a common way to analyze and compare a set of networks is to reduce their complexity by mapping them into a space through the descriptors that characterize them. This is where the problem analyzed in Chapter 3 arises: what is the subset of descriptors that keeps the characteristics of networks as much as possible unchanged in the mapping process, namely projects non-isomorphic networks in different points of the space and groups nearby networks structurally similar and distant networks dissimilar? Through a simulation of networks from four generative models (Random, Scale-free, Small-world and Stochastic block model) and the selection of a wide set of descriptors of the micro, meso and macro-level of network analysis, this Chapter finds evidence of a small subset of descriptors via Subgroup Discovery. This subset is composed by 5 descriptors: the first moment of the Local Clustering Coefficient, 3 Motifs configurations and the descriptor of Smallworldness. The effectiveness of descriptors is evaluated by applying them to the set of binary cultural networks with 60 cultural traits estimated in Chapter 1 and comparing distances between these points-network in the space of the descriptors with popular network distances used in literature. Two are the main innovations: the construction of a new index of cultural distance among countries, in which is included the cultural network of interdependencies among cultural traits; the selection of a small efficient subset of descriptors for mapping in the space of sets of binary networks, which can also be characterized by a different size from each other

Connecting speeds, directions and arrival times of 22 coronal mass ejections from the Sun to 1 AU

Forecasting the in situ properties of coronal mass ejections (CMEs) from remote images is expected to strongly enhance predictions of space weather, and is of general interest for studying the interaction of CMEs with planetary environments. We study the feasibility of using a single heliospheric imager (HI) instrument, imaging the solar wind density from the Sun to 1 AU, for connecting remote images to in situ observations of CMEs. We compare the predictions of speed and arrival time for 22 CMEs (in 2008-2012) to the corresponding interplanetary coronal mass ejection (ICME) parameters at in situ observatories (STEREO PLASTIC/IMPACT, Wind SWE/MFI). The list consists of front- and backsided, slow and fast CMEs (up to $2700 \: km \: s^{-1}$). We track the CMEs to $34.9 \pm 7.1$ degrees elongation from the Sun with J-maps constructed using the SATPLOT tool, resulting in prediction lead times of $-26.4 \pm 15.3$ hours. The geometrical models we use assume different CME front shapes (Fixed-$\Phi$, Harmonic Mean, Self-Similar Expansion), and constant CME speed and direction. We find no significant superiority in the predictive capability of any of the three methods. The absolute difference between predicted and observed ICME arrival times is $8.1 \pm 6.3$ hours ($rms$ value of 10.9h). Speeds are consistent to within $284 \pm 288 \: km \: s^{-1}$. Empirical corrections to the predictions enhance their performance for the arrival times to $6.1 \pm 5.0$ hours ($rms$ value of 7.9h), and for the speeds to $53 \pm 50 \: km \: s^{-1}$. These results are important for Solar Orbiter and a space weather mission positioned away from the Sun-Earth line.Comment: 19 pages, 13 figures, accepted for publication in the Astrophysical Journa

Viral RNA as a branched polymer

Myriad viruses use positive-strand RNA molecules as their genomes. Far from being only a repository of genetic material, viral RNA performs numerous other functions mediated by its physical structure and chemical properties. In this chapter, we focus on its structure and discuss how long RNA molecules can be treated as branched polymers through planar graphs. We describe the major results that can be obtained by this approach, in particular the observation that viral RNA genomes have a characteristic compactness that sets them aside from similar random RNAs. We also discuss how different parameters used in the current RNA folding software influence the resulting structures and how they can be related to experimentally observable quantities. Finally, we show how the connection to branched polymers can be extended to take advantage of known results from polymer physics and can be further moulded to include additional interactions, such as excluded volume or electrostatics.Comment: 24 pages, 9 figure