Based on elementary geometry, a class of novel graph invariants was introduced by Gutman, of which the simplest is the Sombor index SO, defined as italicSO()G=∑italicuv∈EdG2()u+dG2()v, where G = (V, E) is a simple graph and dG(v) denotes the degree of v in G. In this paper, the chemical importance of the Sombor index is investigated and it is shown that the new index is useful in predicting physico‐chemical properties with high accuracy compared to some well‐established and often used indices. We obtain a sharp upper bound for the Sombor index among all molecular trees with fixed numbers of vertices, and characterize those molecular trees achieving the extremal value. Also, we obtain the extremal values of the reduced Sombor index for molecular trees.
The number of dimer-monomers (matchings) of a graph G is an important graph parameter in statistical physics. Following recent research, we study the asymptotic behavior of the number of dimer-monomers m(G) on the Towers of Hanoi graphs and another variation of the Sierpiński graphs which is similar to the Towers of Hanoi graphs, and derive the recursion relations for the numbers of dimer-monomers. Upper and lower bounds for the entropy per site, defined aswhere v(G) is the number of vertices in a graph G, on these Sierpiński graphs are derived in terms of the numbers at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accuracy.
The number of independent sets is equivalent to the partition function of the hardcore lattice gas model with nearest-neighbor exclusion and unit activity. In this article, we mainly study the number of independent sets i(H n) on the Tower of Hanoi graph H n at stage n, and derive the recursion relations for the numbers of independent sets. Upper and lower bounds for the asymptotic growth constant µ on the Towers of Hanoi graphs are derived in terms of the numbers at a certain stage, where µ = lim v→∞ ln i(G) v(G) and v(G) is the number of vertices in a graph G. Furthermore, we also consider the number of independent sets on the Sierpiński graphs which contain the Towers of Hanoi graphs as a special case.
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