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For a family of connected graphs ℱ , a spanning subgraph L of G is called an ℱ -factor if each component of L is isomorphic to a member of ℱ. In this paper, sufficient conditions with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the { P 2 , C 3 , P 5 , (3) } -factor, P ≥ 2 -factor or { P 2 , C 2 i + 1 | i ≥ k } -factor for k ≥ 3 are obtained. [ABSTRACT FROM AUTHOR]

Let G be an edge-colored connected graph. A path P in G is called ℓ -rainbow if no two edges of the same color have fewer than ℓ edges between them on P. The graph G is called (k , ℓ) -rainbow connected if every pair of distinct vertices of G are connected by k pairwise internally vertex-disjoint ℓ -rainbow paths in G. For a k-connected graph G, the minimum number of colors needed to make G (k , ℓ) -rainbow connected is called the (k , ℓ) -rainbow connection number of G and denoted by r c k , ℓ (G) . In this paper, we prove that r c 1 , 2 (G) ≤ 5 for any 2-connected graph G and provide a counter example which overturns a conjecture posed by Chartrand et al. in 2016. Considering graph operations, we study the (1, 2)-rainbow connection numbers for the join, the Cartesian, strong and lexicographic products of graphs, giving the sharp upper bounds for nearly all graph products. In addition, we find some basic properties of the (k , ℓ) -rainbow connection number and determine the values of r c 1 , ℓ (G) where G is a cube or a permutation graph of a nontrivial traceable graph. [ABSTRACT FROM AUTHOR]

The entropy of a graph is an information-theoretic quantity for measuring the complexity of a graph. After Shannon introduced the entropy to information and communication, many generalizations of the entropy measure have been proposed, such as Rényi entropy and Daróczy entropy. In this article, we prove accurate connections (inequalities) between generalized graph entropies, graph energies, and topological indices. Additionally, we obtain some extremal properties of nine generalized graph entropies by employing graph energies and topological indices. [ABSTRACT FROM AUTHOR]

In the work, we prove that γ t (F) ≤ a (F) + 1 holds for a tree-like graph F , where γ t (F) and a (F) denote the total domination number and annihilation number of a connected graph F , respectively. This result generalizes some results of Desormeaux et al. (2013) and Bujtás and Jakovac (2019). [ABSTRACT FROM AUTHOR]