Harmonic–Arithmetic Index for the Generalized Mycielskian Graphs and Graphenes With Curvilinear Regression Models of Benzenoid Hydrocarbons

This is a summary of the author’s Ph.D. thesis supervised by Sara Nicoloso and Gianpaolo Oriolo and defended on 3 April 2008 at Sapienza Universita di Roma. The thesis is written in English and is available from the author upon request. This work deals with three classical combinatorial problems, namely the isomorphism, the vertex-coloring and the stable set problem, restricted to two graph classes, namely circulant and claw-free graphs. In the first part (joint work with Sara Nicoloso), we derive a necessary and sufficient condition to test isomorphism of circulant graphs, and give simple algorithms to solve the vertex-coloring problem on this class of graphs. In the second part (joint work with Gianpaolo Oriolo and Gautier Stauffer), we propose a new combinatorial algorithm for the maximum weighted stable set problem in claw-free graphs, and devise a robust algorithm for the same problem in the subclass of fuzzy circular interval graphs, which also provides recognition when the stability number is greater than three.

A linear description of the stable set polytope STAB(G) of a quasi-line graph G is given in Eisenbrand et al. (Combinatorica 28(1):45–67, 2008), where the so called Ben Rebea Theorem (Oriolo in Discrete Appl Math 132(3):185–201, 2003) is proved. Such a theorem establishes that, for quasi-line graphs, STAB(G) is completely described by non-negativity constraints, clique inequalities, and clique family inequalities (CFIs). As quasi-line graphs are a superclass of line graphs, Ben Rebea Theorem can be seen as a generalization of Edmonds’ characterization of the matching polytope (Edmonds in J Res Natl Bureau Stand B 69:125–130, 1965), showing that the matching polytope can be described by non-negativity constraints, degree constraints and odd-set inequalities. Unfortunately, the description given by the Ben Rebea Theorem is not minimal, i.e., it is not known which are the (non-rank) clique family inequalities that are facet defining for STAB(G). To the contrary, it would be highly desirable to have a minimal description of STAB(G), pairing that of Edmonds and Pulleyblank (in: Berge, Chuadhuri (eds) Hypergraph seminar, pp 214–242, 1974) for the matching polytope. In this paper, we start the investigation of a minimal linear description for the stable set polytope of quasi-line graphs. We focus on circular interval graphs, a subclass of quasi-line graphs that is central in the proof of the Ben Rebea Theorem. For this class of graphs, we move an important step forward, showing some strong sufficient conditions for a CFI to induce a facet of STAB(G). In particular, such conditions come out to be related to the existence of certain proper circulant graphs as subgraphs of G. These results allows us to settle two conjectures on the structure of facet defining inequalities of the stable set polytope of circulant graphs (Pêcher and Wagler in Math Program 105:311–328, 2006) and of (fuzzy) circular graphs (Oriolo and Stauffer in Math Program 115:291–317, 2008), and to slightly refine the Ben Rebea Theorem itself.

In the present thesis we investigate algebraic and arithmetic properties of graph spectra. In particular, we study the algebraic degree of a graph, that is the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals, and examine the question whether there is a relation between the algebraic degree of a graph and its structural properties. This generalizes the yet open question ``Which graphs have integral spectra?'' stated by Harary and Schwenk in 1974. We provide an overview of graph products since they are useful to study graph spectra and, in particular, to construct families of integral graphs. Moreover, we present a relation between the diameter, the maximum vertex degree and the algebraic degree of a graph, and construct a potential family of graphs of maximum algebraic degree. Furthermore, we determine precisely the algebraic degree of circulant graphs and find new criteria for isospectrality of circulant graphs. Moreover, we solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Those results generalize a theorem of So who characterized all integral circulant graphs. For our proofs we exploit the theory of Schur rings which was already used in order to solve the isomorphism problem for circulant graphs. Besides that, we study spectra of zero-divisor graphs over finite commutative rings. Given a ring \(R\), the zero-divisor graph over \(R\) is defined as the graph with vertex set being the set of non-zero zero-divisors of \(R\) where two vertices \(x,y\) are adjacent if and only if \(xy=0\). We investigate relations between the eigenvalues of a zero-divisor graph, its structural properties and the algebraic properties of the respective ring.

Several interconnection networks (such as rings, meshes and hypercubes) can be modeled as circulant graphs. As a result, methods previously developed for constructing fault-tolerant solutions of circulant graphs can also be applied to these networks. Among these methods, the one based on the idea of offsets partitioning is the most efficient (for circulant graphs). We review this method in this paper, and extend its applications to hypercubes. Moreover, we develop new algorithms to reconfigure circulant graphs and hypercubes. Our results show that the fault-tolerant solutions obtained, and the reconfiguration algorithms developed are efficient.

Several interconnection networks (such as meshes and hypercubes) can be modeled as circulant graphs. As a result, methods previously developed for constructing fault-tolerant solutions of circulant graphs can also be applied to these networks. Among these methods, the one based on the idea of "offsets partitioning" is the most efficient (for circulant graphs). We review this method in this paper, and extend its applications to hypercubes. Moreover, we develop new algorithms to reconfigure circulant graphs and hypercubes. Our results show that the fault-tolerant solutions obtained, and the reconfiguration algorithms developed are efficient.

Let G be a nontrivial connected graph with vertex set V(G). For an ordered set W = {w1, w2, …, wn} of n distinct vertices in G, the representation of a vertex v ∈ V(G) with respect to W is an ordered value of distance between v and every vertex of W . The set W is a local metric set of G if the representations of every pair of adjacent vertices with respect to W are different. The local metric set with minimum cardinality is called local metric basis and its cardinality is the local metric dimension of G and denoted by diml(G). The edge corona product of cycle graph and path graph denoted by Cm ʘ Pn, this graph is obtained from a cycle graph Cm and m copies of path graph Pn, and then joining two end-vertices of ith edge of Cm to every vertex in the ith copy of Pn, where 1 ≤ i ≤ m. The corona product of cycle graph and path graph denoted by Cm ◊ Pn, this graph is obtained from a cycle graph Cm and m copies of path graph Pn, and then joining by an edge each vertex from the ith copy of Pn with ith vertex of Cm. In this paper, we determine the local metric dimension of edge corona and corona product of cycle graph and path graph for positive integer m ≥ 3 and n ≥ 1. We obtain the local metric dimension of edge corona product of cycle and path graphs is diml(Cm ◊ Pn)=2 for n = 1 and for n ≥ 2. The local metric dimension of corona product of cycle and path graphs is diml(Cm ◊ Pn)=1 for n = 1 and even positive integer m ≥ 3, diml(Cm ʘ Pn)=2 for n = 1 and odd positive integer m ≥ 3, and for n ≥ 2.

Integrating structural and seismic properties for enhanced seismic response prediction of building structures via artificial neural network

Objectives: To compute the packing chromatic number of transformation of path graph, cycle graph and wheel graph. Methods: The packing chromatic number of Xpc (H) of a graph H is the least integer m in such a way that there is a mapping C: V(H)→(1,2,…,m} such that the distance between any two nodes of colour k is greater than k+1. Findings: The packing chromatic number of the transformation of the graph Xpc (Hpqr) where p,q,r be variables which has the values either positive sign (+)+ or a negative sign (-) then Hpqr is known as the transformation of the graph H such that VH and E(H) belonging to the vertex set of Hpqr and α(H), β(H) also belonging to V(H), E(H) of the graph. Obtained the values of the packing chromatic number of transformation of path graph, cycle graph and wheel graph. Applications: Chromatic number applied in Register Allocations, a compiler is a computer program that translates one computer language into another. To improve the execution time of the resulting code, one of the techniques of compiler optimization is register allocation; if the graph can be colored with k colors then the variables can be stored in k registers. Keywords path graph, cycle graph, wheel graph, packing chromatic number

In this thesis we focus our attention on the stable set polytope of claw-free graphs. This problem has been open for many years and albeit all the efforts engaged during those last three years, it is still open. This does not mean that no progress has been made and we hope to the contrary that this thesis contains some important advances and that the reader will share this point of view. Understanding the stable set polytope requires seizing its geometry. But its most interesting facets appear in high dimension and thus one must develop a more algebraic than geometric intuition of the problem. In this thesis we try to convey the intuition we have developed. Indeed we provide several constructions and examples of facets for the stable set polytope of circulant graphs, quasi-line graphs and claw-free graphs. This intuition allowed us to prove some important results in the characterization of the stable set polytope. The main contribution of the thesis is without any doubt the proof of Ben Rebea conjecture leading to a characterization of the stable set polytope of quasi-line graphs. This result claims that clique, non-negativity and clique family inequalities are enough to characterize this polytope. It builds upon a recent decomposition theorem by Chudnovsky and Seymour for claw-free graphs. This is not the end of the story and we try to determine which of the inequalities are essential i.e. which are the facets of the polytope. We answer this question in various ways. First we propose a combinatorial characterization of the rank facets of the stable set polytope of fuzzy circular interval graphs. Then we prove that all the facets of the stable set polytope of quasi-line graphs can be obtained by lifting rank facets in lower dimension and thus we determine a strongly minimal characterization of the facets of this polytope. Finally we give strong necessary conditions for clique family inequalities to be facet producing in quasi-line graphs. It allows in particular to give a better characterization of the stable set polytope of circular interval graphs which proves a conjecture by Pecher and Wagler for the stable set polytope of circulants. Even if we mainly focus on polyhedral results, we also derive various algorithmic ones. We develop an algorithm to solve the weighted stable set problem in fuzzy circular interval graphs and another one to determine a maximum cardinality stable set in a quasi-line graph. We finally provide procedures to separate over certain valid inequalities for the stable set polytope of general graphs. We conclude the thesis with some perspectives. In particular, we formulate several conjectures about the stable set polytope of claw-free graphs and some possible lines of attack. We hope this will help researchers interested in continuing those works.

A successful approach to the development of reinforced materials for enhanced cutting tool inserts requires the formulation and application of innovative concepts at each step of material design development. In this paper, reinforced ceramic-based cutting tools with enhanced thermal and structural properties are developed for high-speed machining applications using a computational approach. A mean-field homogenization, effective medium approximation and J-integral based fracture toughness evaluation using an in-house code are used for predicting the effective structural and thermal properties for tool inserts as a function of reinforcement type, volume fraction, particle size and interface between matrix and reinforcement. Initially, several potential reinforcements are selected at the material design phase. SiC, TiB2, cBN and TiC were all found to be suitable candidates when reinforced into an alumina matrix as both single and hybrid inclusions for the enhancement of thermal and structural properties. For validation purposes, alumina-cubic boron nitride-silicon carbide composites are developed using Spark Plasma Sintering as hybrid systems, which are in line with the designed range of volume fraction and reinforcement particle size. Structural and thermal properties such as elastic modulus, fracture toughness and thermal conductivity are measured to complement the computational material design model.

BackgroundAmyloid-beta (Aβ) has a dose-response relationship with cognition in healthy adults. Additionally, the levels of functional connectivity within and between brain networks have been associated with cognitive performance in healthy adults. Aiming to explore potential synergistic effects, we investigated the relationship of inter-network functional connectivity, Aβ burden, and memory decline among healthy individuals and individuals with preclinical, prodromal, or clinical Alzheimer’s disease.MethodsIn this longitudinal cohort study (ADNI2), participants (55–88 years) were followed for a maximum of 5 years. We included cognitively healthy participants and patients with mild cognitive impairment (with or without elevated Aβ) or Alzheimer’s disease. Associations between memory decline, Aβ burden, and connectivity between networks across the groups were investigated using linear and curvilinear mixed-effects models.ResultsWe found a synergistic relationships between inter-network functional connectivity and Aβ burden on memory decline. Dose-response relationships between Aβ and memory decline varied as a function of directionality of inter-network connectivity across groups. When inter-network correlations were negative, the curvilinear mixed-effects models revealed that higher Aβ burden was associated with greater memory decline in cognitively normal participants, but when inter-network correlations were positive, there was no association between the magnitude of Aβ burden and memory decline. Opposite patterns were observed in patients with mild cognitive impairment. Combining negative inter-network correlations with Aβ burden can reduce the required sample size by 88% for clinical trials aiming to slow down memory decline.ConclusionsThe direction of inter-network connectivity provides additional information about Aβ burden on the rate of expected memory decline, especially in the preclinical phase. These results may be valuable for optimizing patient selection and decreasing study times to assess efficacy in clinical trials.

Wirelength of embedding complete multipartite graphs into certain graphs

Objectives: Introducing the idea of energy pebbling to path, star, tree, and cycle graphs is the main objective of this study. The sum of the absolute values of the eigenvalues in the adjacency matrix of a pebbling network is its energy. We then construct the upper and lower bounds for the aforementioned graphs in this discussion. Methods: Removing two pebbles from the first vertex and placing one pebble on the neighboring vertex illustrates a pebbling move. The pebbling graph's adjacency matrix is computed as where is the edge value of is adjacent to . The energy of the graph is calculated using , where are the eigenvalues of the graph G. Analyses are made for the cycle, path, star, and tree graphs' lower and upper bounds. Findings: In regards to the path, star, cycle, and tree pebbling graph, the relationship between energy, and lower and upper bounds was discovered and tabulated. Novelty: Pebbling path, star, cycle, and tree graphs were subjected to the energy idea, and a relationship was found between the lower bound, upper bound, and energy of pebbling graphs in general. Keywords: Pebbling, Adjacency energy, Lower bound, Energy

For a graph G, let b(G)=max﹛|D|: Dis an edge cut of G﹜ . For graphs G and H, a map Ψ: V(G)→V(H) is a graph homomorphism if for each e=uv∈E(G), Ψ(u)Ψ(v)∈E(H). In 1979, Erdos proved by probabilistic methods that for p ≥ 2 with if there is a graph homomorphism from G onto Kp then b(G)≥f(p)|E(G)| In this paper, we obtained the best possible lower bounds of b(G) for graphs G with a graph homomorphism onto a Kneser graph or a circulant graph and we characterized the graphs G reaching the lower bounds when G is an edge maximal graph with a graph homomorphism onto a complete graph, or onto an odd cycle.

The study of thermodynamics and electronic structure of chemotherapy drug is crucial in developing effective cancer treatments. Quantitative Structure-Property Relationship (QSPR) analysis is an essential instrument in creating and enhancing chemotherapeutic drugs. This research employs Density Functional Theory (DFT) to compute thermodynamical and electronic characteristics of different chemotherapeutic drugs. Distance-based topological descriptors are utilized to assess the molecular structure of these chemotherapy drugs. These descriptors are subsequently employed in curvilinear regression models to forecast essential thermodynamical attributes and biological activities. We seek to improve the precision of QSPR models by correlating DFT-derived attributes with topological descriptors via curvilinear regression methods. Our results indicate that curvilinear regression models, especially those with quadratic and cubic curve fitting, markedly enhance the prediction capability for analyzing thermodynamical properties of drugs. Our findings further specify that Wiener index and Gutman index outperformed the indices in predicting the properties of drugs. This method offers an enhanced understanding of the thermodynamics of chemotherapeutic medicines and promotes the creation of more effective and safer therapeutic compounds. The findings could pave the way for more precise and personalised cancer treatment strategies, ultimately improving patient outcomes. The application of topological indices in QSPR modelling, which accounts for molecular symmetry, has significant promise in enhancing our comprehension of compounds' structural and thermodynamical characteristics.