On connected graphs and trees with maximal inverse sum indeg index

Let <em>G </em>= (<em>V</em>,<em>E</em>) be a simple connected graph with the vertex set <em>V</em> = {1,2,...,<em>n</em>} and sequence of vertex degrees (<em>d</em>₁,<em>d</em>₂,...,<em>d_n</em>) where <em>d_i</em> denotes the degree of a vertex <em>i</em> ∈ <em>V</em>. With <em>i</em> ∼ <em>j</em>, we denote the adjacency of the vertices <em>i</em> and <em>j</em> in the graph <em>G</em>. The inverse sum indeg (<em>ISI</em>) index of the graph <em>G</em> is defined as <em>ISI</em>(<em>G</em>)=∑_{<em>i</em>∼<em>j</em>}(<em>d_id_j</em>)/(<em>d_i</em>+<em>d_j</em>). Some new upper bounds for the <em>ISI</em> index are obtained in this paper.

Extremal values on the eccentric distance sum of trees

On the inverse sum indeg index

Objective: Molecular topology, chemical-based graph theory, and mathematical chemistry all depend on the study of topological criteria because these metrics capture the structural, chemical, and physical properties of molecules. However, there is a clear challenge in characterising non-isomorphic graphs that have the same topological measure value. Numerous areas of chemistry benefit from the application of algebraic graph theory. It is a strong theoretical method for predicting either the common and unusual properties of molecules because it clarifies how the many orientations of macromolecules and crystals impact their structure and behaviour.Methods : In this study, we first calculate the M and NM polynomials of the graph and then recover some degree-based and neighbourhood degree sum based indices. In addition, a few outcomes are shown graphically for contrast. Findings : First Zagreb Index (U_1), Second Zagreb Index (U_2), Redefined third Zagreb Index (U_3), Y-Tally (U_y), Forgotten topological Index (U_f ), Second modified Zagreb Index (U_m), Symmetric division deg Index (U_d ), Harmonic Index (U_h), Inverse sum indeg Index (U_i), and Augmented Zagreb Index (U_a) are among the topographies of the Graph that this work aims to forecast and analyse using fuzzy mathematical modelling. Novelty: This research used ANOVA one-way classification to show that the graph's various topographies are the same.

In this paper, we study the edge version of some degree based topological indices such as general sum-connectivity index, Randic index, inverse sum indeg index, symmetric division deg index, augmenting Zagreb index and harmonic polynomial for joint graphs and certain graph operations.

The quantitative specifications which are used to narrate the atomic topology of graphs are usually entitled as topological indices in theoretical Chemistry. The features of physical and chemical properties like melting point, entropy, boiling point, energy generation and enthalpy of vaporization of chemical compounds can be estimated by means of these topological indices. The field of Graph theory has a remarkable application of linking certain graphs with many types of topological indices. In this paper we compute first and second Zagreb indices, Randi´c index, sum-connectivity index, harmonic index, inverse sum indeg index, modified first and second Zagreb indices and first and second hyper Zagreb indices of Triangular snake graph and alternate Triangular snake graph.

Following work of Stroud and Saeger (Proceedings of ISI, Springer Verlag, New York, 2006) and Anand et al. (Proceedings of Computer, Communication and Control Technologies, 2003), we formulate a port of entry inspection sequencing task as a problem of finding an optimal binary decision tree for an appropriate Boolean decision function. We report on new algorithms for finding such optimal trees that are more efficient computationally than those presented by Stroud and Saeger and Anand et al. We achieve these efficiencies through a combination of specific numerical methods for finding optimal thresholds for sensor functions and two novel binary decision tree search algorithms that operate on a space of potentially acceptable binary decision trees. The improvements enable us to analyze substantially larger applications than was previously possible. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

When doing a bootstrap analysis with a single tree saved per pseudoreplicate, biased search algorithms may influence support values more than actual properties of the data set. Two methods commonly used for finding phylogenetic trees consist of randomizing the input order of species in multiple addition sequences followed by branch swapping, or using random trees as the starting point for branch swapping. The randomness inherent to such methods is assumed to eliminate any consistent preferences for some trees or unsupported groups of taxa, but both methods can be significantly biased. In the case of trees created by sequentially adding taxa, a bias may occur even if every addition sequence is equiprobable, and if one of the equally optimal positions for each terminal to add to the tree is selected equiprobably. In the case of branch swapping, the bias can happen even when branch swapping equiprobably selects any of the trees of better score in the subtree-pruning-regrafting-neighborhood or tree-bisection-reconnection-neighborhood. Consequently, when the data set is ambiguous, both random-addition sequences and branch swapping from random trees may (i) find some of the optimal trees much more frequently than others and (ii) find some groups with a frequency that differs from their frequency among all optimal trees. When the data set defines a single optimal tree, the groups present in that tree may have a different probability of being found by a search, even if supported by equal amounts of evidence. This may happen in both parsimony and maximum-likelihood analyses, and even in small data sets without incongruence.

For a tree T , let l p ( T ) be the number of different lengths of leaf to leaf paths in T . For a degree sequence s of a tree, let rad ( s ) be the minimum radius of a tree with degree sequence s . Recently, Di Braccio, Katsamaktsis, Ma, Malekshahian, and Zhao provided a lower bound on l p ( T ) in terms of the number of leaves and the maximum degree of T , answering a related question posed by Narins, Pokrovskiy, and Szabó. Here we show l p ( T ) ≥ rad ( s ) − log 2 ⁡ ( rad ( s ) ) for a tree T with no vertex of degree 2 and degree sequence s , and discuss possible improvements and variants.

We describe algorithms for constructing optimal binary search trees, in which the access cost of a key depends on the k preceding keys, which were reached in the path to it. This problem has applications to searching on secondary memory and robotics. Two kinds of optimal trees are considered, namely optimal worst case trees and weighted average case trees. The time and space complexity of both algorithms are O(n k+2) and O(n k+1), respectively. The algorithms are based on a convenient decomposition and characterizations of sequences of keys, which are paths of special kinds in binary search trees. Finally, using generating functions, the exact number of steps performed by the algorithms has been calculated. The subject will be introduced by a general discussion on the construction of optimal binary search trees.

We study a scale‐free random graph process in which the number of edges added at each step increases. This differs from the standard model in which a fixed number,m, of edges are added at each step.Letf(t) be the number of edges added at stept. In the standard scale‐free model,f(t) =mconstant, whereas in this paper we considerf(t) = [tc],c&gt; 0. Such a graph process, in which the number of edges grows non‐linearly with the number of vertices is said to have accelerating growth.We analyze both an undirected and a directed process. The power law of the degree sequence of these processes exhibits widely differing behavior.For the undirected process, the terminal vertex of each edge is chosen by preferential attachment based on vertex degree. Whenf(t) =mconstant, this is the standard scale‐free model, and the power law of the degree sequence is 3. Whenf(t) = [tc],c&lt; 1, the degree sequence of the process exhibits a power law with parameterx= (3 −c)/(1 −c). Asc→ 0,x→ 3, which gives a value ofx= 3, as in standard scale‐free model. Thus no more slowly growing monotone functionf(t) alters the power law of this model away fromx= 3. Whenc= 1, so thatf(t) =t, the expected degree of all vertices ist, the vertex degree is concentrated, and the degree sequence does not have a power law.For the directed process, the terminal vertex is chosen proportional to in‐degree plus an additive constant, to allow the selection of vertices of in‐degree zero. For this process whenf(t) =mis constant, the power law of the degree sequence isx= 2 + 1/m. Whenf(t) = [tc],c&gt; 0, the power law becomesx= 1 + 1/(1 +c), which naturally extends the power law to [1,2]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 396–421, 2011

Optimal binary search trees with costs depending on the access paths

A general critical condition for the emergence of a giant component in random graphs with given degrees

Trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence are characterized. It is shown that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p.

The notion of hypergraph generalizes that of graph in the sense that each hyperedge is a non-void subset of the set of vertices, without constraints on its cardinality. A fundamental and widely investigated notion related both to graphs and to hypergraphs is the characterization of their degree sequences, that is the lists of their vertex degrees. Concerning graphs, this problem has been solved in a classical study by Erdi¾?s and Gallai, while no efficient solutions are known for hypergraphs. If we restrict the degree sequences characterization to uniform hypergraphs, several necessary conditions are provided in the literature, but only few sufficient ones: among the latter, a recent one requires to split a sequence into suitable subsequences whose graphicality has to be recursively tested. Unfortunately, such an approach does not allow a direct efficient implementation. We study this problem under a tomographical perspective by adapting an already known reconstruction algorithm that has been defined for regular h-uniform degree sequences to the proposed instances, providing efficiency to the sufficient condition. Furthermore, we extend the set of h-uniform degree sequences whose graphicality can be efficiently tested. This tomographical approach seems extremely promising for further developments.