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For a graph G = (V , E) and a set S ⊆ V (G) of size at least 2 , a path in G is said to be an S -path if it connects all vertices of S. Two S -paths P 1 and P 2 are said to be internally disjoint if E (P 1) ∩ E (P 2) = ∅ and V (P 1) ∩ V (P 2) = S. Let π G (S) denote the maximum number of internally disjoint S -paths in G. The k -path-connectivity π k (G) of G is then defined as the minimum π G (S) , where S ranges over all k -subsets of V (G). In [M. Hager, Path-connectivity in graphs, Discrete Math. 59 (1986) 53–59], the k -path-connectivity of the complete bipartite graph K a , b was calculated, where k ≥ 2. But, from his proof, only the case that 2 ≤ k ≤ min { a , b } was considered. In this paper, we calculate the situation that min { a , b } + 1 ≤ k ≤ a + b and complete the result. [ABSTRACT FROM AUTHOR]

The g -extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464–473] determined the 3 -extra edge-connectivity of balanced hypercubes B H n and conjectured that the g -extra edge-connectivity of B H n is λ g ( B H n) = 2 (g + 1) n − 4 g + 4 for 2 ≤ g ≤ 2 n − 1. In this paper, we confirm their conjecture for n ≥ 6 − 1 2 g + 1 and 2 ≤ g ≤ 8 , and disprove their conjecture for n ≥ 3 e g ( B H n) g + 1 and 9 ≤ g ≤ 2 n − 1 , where e g ( B H n) = max { | E ( B H n [ U ]) | | U ⊆ V (B H n) , | U | = g + 1 }. [ABSTRACT FROM AUTHOR]

The Multi-Tier (M-Tier) algorithm proposed in [1], has enabled an enhanced correlation between dynamic resource utilization and QoS guarantees. The algorithm favored the delay sensitive applications. Indirectly, this infers that the assignment of priorities practiced in the algorithm is static. In redesigning the algorithm, priority assignments should be able to be implemented dynamically. Thus, enabling an embedded element to provide implementation mobility. This would depend upon the current status of the classified buffers and the tolerance level of the delay sensitive traffic. Integrating control parameters to enable alternation in scheduling priorities is an option to resolve the element of partiality present in the M-Tier algorithm. Thereafter, the proposed algorithm will be denoted as the M-Tier(T) algorithm [2]. The paper proposes a detailed mathematical model for the analysis of dynamic resource reservation algorithms implemented in IP switches. The semi-markov process (SMP) approach utilized provides an accurate performance prediction and representation of resource reservation mechanisms. The algorithm correlates the average packet delay and packet loss ratio parameters to activate the dynamic resource reservation mechanism in a hierarchical manner. The developed models are validated through discrete-event simulation. The performance of the algorithm is studied in terms of average packet delay and network throughput. [ABSTRACT FROM AUTHOR]

The triple-loop network associated with the hyper-L shape has been shown to have the best diameter among all triple-loop networks so far. In this paper we study the k- diameters of this network. In particular, we construct three node-disjoint shortest paths no longer than the diameter plus 2 for any pair of nodes. [ABSTRACT FROM AUTHOR]