On the topological index

The computational complexity of the Hosoya index of a given graph is NP-Complete. Let RT(G) be the graph constructed from R(G) by a triangle instead of all vertices of the initial graph G. In this ...

The hosoya index of graphs formed by a fractal graph

For a given simple graph G, the energy of G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix, which was defined by I. Gutman. The problem on determining the maximal energy tends to be complicated for a given class of graphs. There are many approaches on the maximal energy of trees, unicyclic graphs and bicyclic graphs, respectively. Let P 6,6,6 n denote the graph with n ≥ 20 vertices

On a Conjecture about Tricyclic Graphs with Maximal Energy

Let G be a simple, connected graph of order n with eigenvalues λ1 ≥ λ2 ≥ · ·· ≥ λn . The median eigenvalues λH ,λ L ,f orH = �(n +1 )/2� and L = �(n +1 )/2� are considered. These play an important

Note on the HOMO-LUMO Index of Graphs ∗

Abstract The Wiener index W(G) of a connected graph G, introduced by Wiener in 1947, is defined as W(G) =∑u,v∈V (G) dG(u, v), where dG(u, v) is the distance (the length a shortest path) between the vertices u and v in G. For S ⊆ V (G), the Steiner distance d(S) of the vertices of S, introduced by Chartrand et al. in 1989, is the minimum size of a connected subgraph of G whose vertex set contains S. The k-th Steiner Wiener index SWk(G) of G is defined as SWk(G)=∑S⊆V(G)|S|=kd(S) $SW_k (G) = \sum\nolimits_{\mathop {S \subseteq V(G)}\limits_{|S| = k} } {d(S)}$ . We investigate the following problem: Fixed a positive integer k, for what kind of positive integer w does there exist a connected graph G (or a tree T) of order n ≥ k such that SWk(G) = w (or SWk(T) = w)? In this paper, we give some solutions to this problem.

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Inverse Problem on the Steiner Wiener Index

Greedy trees are constructed from a given degree sequence by a simple greedy algorithm that assigns the highest degree to the root, the second-, third-, ... highest degrees to the root's neighbors, and so on. They have been shown to maximize or minimize a number of different graph invariants among trees with a given degree sequence. In particular, the total number of subtrees of a tree is maximized by the greedy tree. In this work, we show that in fact a much stronger statement holds true: greedy trees maximize the number of subtrees of any given order. This parallels recent results on distance-based graph invariants. We obtain a number of corollaries from this fact and also prove analogous results for related invariants, most notably the number of antichains of given cardinality in a rooted tree.

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Greedy trees, subtrees and antichains

Unicyclic graphs with large energy

Energy, Hosoya index and Merrifield-Simmons index of trees with prescribed degree sequence

A segment of a tree is a path whose ends are branching vertices (vertices of degree greater than 2) or leaves, while all other vertices have degree 2. The lengths of all the segments of a tree form its segment sequence. In this note we consider the problem of maximizing the Wiener index among trees with given segment sequence or number of segments, answering two questions proposed in a recent paper on the subject. We show that the maximum is always obtained for a so-called quasi-caterpillar, and we also further characterize its structure. ∗This work was supported by grants from the Simons Foundation (#245307) and the National Research Foundation of South Africa (grants 96236 and 96310). MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 75 (2016) 91-104

Maximum Wiener Index of Trees With Given Segment Sequence

Let $\lambda_1,\dots,\lambda_n$ be the eigenvalues of a graph $G$. For any $k\geq 0$, the $k$-th spectral moment of $G$ is defined by $\M_k(G)=\lambda_1^k+\dots+\lambda_n^k$. We use the fact that $\M_k(G)$ is also the number of closed walks of length $k$ in $G$ to show that among trees $T$ whose degree sequence is $D$ or majorized by $D$, $\M_k(T)$ is maximized by the greedy tree with degree sequence $D$ (constructed by assigning the highest degree in $D$ to the root, the second-, third-, \dots highest degrees to the neighbors of the root, and so on) for any $k\geq 0$. Several corollaries follow, in particular a conjecture of Ilic and Stevanovic on trees with given maximum degree, which in turn implies a conjecture of Gutman, Furtula, Markovic and Glisic on the Estrada index of such trees, which is defined as $\EE(G)=e^{\lambda_1}+\dots+e^{\lambda_n}$.

Eric Ould Dadah Andriantiana

Papers

Greedy trees, subtrees and antichains

Unicyclic graphs with large energy

Energy, Hosoya index and Merrifield-Simmons index of trees with prescribed degree sequence

Maximum Wiener Index of Trees With Given Segment Sequence

Spectral moments of trees with given degree sequence