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In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

In this paper, we show the following: Let G be a connected claw-free graph such that G has a connected induced subgraph H that has a pair of vertices { v 1 , v 2 } of degree one in H whose distance is d + 2 in H. Then H has an induced subgraph F, which is isomorphic to B i , j , with { v 1 , v 2 } ⊆ V (F) and i + j = d + 1 , with a well-defined exception. Here B i , j denotes the graph obtained by attaching two vertex-disjoint paths of lengths i , j ≥ 1 to a triangle. We also use the result above to strengthen the results in Xiong et al. (Discrete Math 313:784–795, 2013) in two cases, when i + j ≤ 9 , and when the graph is Γ 0 -free. Here Γ 0 is the simple graph with degree sequence 4, 2, 2, 2, 2. Let i , j > 0 be integers such that i + j ≤ 9 . Then every 3-connected { K 1 , 3 , B i , j } -free graph G is hamiltonian, and every 3-connected { K 1 , 3 , Γ 0 , B 2 i , 2 j } -free graph G is hamiltonian. The two results above are all sharp in the sense that the condition " i + j ≤ 9 " couldn't be replaced by ' ' i + j ≤ 10 ". [ABSTRACT FROM AUTHOR]

This paper investigates the entanglement dynamics of an anisotropic two-qubit Heisenberg spin chain in the presence of decoherence at finite temperature. The time evolution of the concurrence is studied for different initial Werner states. The influences of initial purity, finite temperature, spontaneous decay and Hamiltonian on the entanglement evolution are analyzed in detail. Our calculations show that the finite temperature restricts the evolution of the entanglement all the time when the Hamiltonian improves it and the spontaneous decay to the reservoirs can produce quantum entanglement with the anisotropy of spin-spin interaction. Finally, the steady-state concurrence which may remain non-zero for low temperature is also given. [ABSTRACT FROM AUTHOR]

A graph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a Hamiltonian graph and let x and y be vertices of G that are consecutive on some Hamiltonian cycle in G. Hakimi and Schmeichel showed (J Combin Theory Ser B 45:99-107, 1988) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n - 1 or G is isomorphic to a complete bipartite graph. In this paper, we study the existence of cycles of various lengths in a Hamiltonian graph G given the existence of a pair of vertices that have a high degree sum but are not adjacent on any Hamiltonian cycle in G. [ABSTRACT FROM AUTHOR]

It is conjectured that every fullerene graph is hamiltonian. Jendrol’ and Owens proved (J. Math. Chem. 18:83–90, 1995) that every fullerene graph on n vertices has a cycle of length at least 4 n/5. Recently, Král’ et al. improved it to 5 n/6 − 2/3. In this paper, we study 2-factors of fullerene graphs. As a by-product, we get an improvement of a lower bound on the length of the longest cycle in a fullerene graph. We present a constructive proof of the bound 6 n/7 + 2/7. [ABSTRACT FROM AUTHOR]

Let G be a graph and S ⊂ V( G). We denote by α( S) the maximum number of pairwise nonadjacent vertices in S. For x, y ∈ V( G), the local connectivity κ( x, y) is defined to be the maximum number of internally-disjoint paths connecting x and y in G. We define $$\kappa(S)=\min\{\kappa(x,y) : x,y \in S,x\not=y\}$$ . In this paper, we show that if κ( S) ≥ 3 and $$\sum_{i=1}^4 d_{G}{(x_i)} \ge |V(G)|+\kappa(S)+\alpha (S)-1$$ for every independent set { x 1, x 2, x 3, x 4} ⊂ S, then G contains a cycle passing through S. This degree condition is sharp and this gives a new degree sum condition for a 3-connected graph to be hamiltonian. [ABSTRACT FROM AUTHOR]

A graph G is quasi-claw-free if it satisfies the property: d(x, y)=2 ⇒ there exists u ε N (x) ∩ N (y) such that N[u] ⊆ N[x] ∪ N[y]. In the paper, we prove that the circumference of a 3-connected quasi-claw-free graph G on n vertices is at least min{4δ - 2, n} and G is hamiltonian if n ≤ 5δ - 5. [ABSTRACT FROM AUTHOR]

In this paper, we completely determine the connectivity of every infinite circulant digraphs and prove that almost all infinite circulant digraphs are infinitely strongly connected and therefore have both one- and two-way infinite Hamiltonian paths. [ABSTRACT FROM AUTHOR]