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One of the symbolic parameters to measure the fault tolerance of a network is its connectivity. The HH-structure connectivity and HH-substructure connectivity extend the classical connectivity and are more practical. For a graph GG and its connected subgraph HH, the HH-structure connectivity κ(G;H)\kappa \left(G;\hspace{0.33em}H) (resp. HH-substructure connectivity κs(G;H){\kappa }^{s}\left(G;\hspace{0.33em}H)) of GG is the cardinality of a minimum subgraph set such that every element of the set is isomorphic to HH (resp. every element of the set is isomorphic to a connected subgraph of HH) in GG, whose vertices removal disconnects GG. In this article, we investigate the HH-structure connectivity and HH-substructure connectivity of the nn-dimensional burnt pancake network BPn{{\rm{BP}}}_{n} for each H∈{K1,K1,1,…,K1,n−1,P4,…,P7,C8}H\in \left\{{K}_{1},{K}_{1,1},\ldots ,{K}_{1,n-1},{P}_{4},\ldots ,{P}_{7},{C}_{8}\right\}.

The g-extra connectivity is a very important index to evaluate the fault tolerance, reliability of interconnection networks. Let g be a non-negative integer, G be a connected graph with vertex set V and edge set E, a subset S ⊆ V is called a g-extra cut of G if the graph induced by the set G − S is disconnected and each component of G − S has at least g + 1 vertices. The g-extra connectivity of G, denoted as κ g (G) , is the cardinality of the minimum g-extra cut of G. Mycielski introduced a graph transformation to discover chromatic numbers of triangle-free graphs that can be arbitrarily large. This transformation converts a graph G into a new compound graph called μ (G) , also known as the Mycielskian graph of G. In this paper, we study the relationship on g-extra connectivity between the Mycielskian graph μ (G) and the graph G. In addition, we show that κ 3 (μ (G)) = 2 κ 1 (G) + 1 for κ 1 (G) ≤ m i n { 4 , ⌊ n 2 ⌋ } , and prove the bounds of κ 2 g + 1 (μ (G)) for g ≥ 2 . [ABSTRACT FROM AUTHOR]

The matching preclusion number of graph G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or almost-perfect matching. The strong matching preclusion number of a graph G is the minimum number of vertices and edges whose deletion leaves the resulting graph without a perfect matching or an almost-perfect matching. The conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or almost-perfect matching. In this paper, we study the matching preclusion number of radix triangular mesh with an odd number of vertices, and strong matching preclusion number and conditional matching preclusion number of radix triangular mesh. Also, we obtained the radix triangular mesh with an even number of vertices is super strongly matched and conditionally super matched. [ABSTRACT FROM AUTHOR]

In this paper, using the properties of chromatic polynomial and adjoint polynomial, we characterize all graphs having chromatic polynomial ∑l⩽m0l/m0−l(λ)l. [Copyright &y& Elsevier]

The improved engineering properties of the expansive soil by mixing with various additives will be changed during the long-term variation of the meteorological and hydrological conditions. In the present work, a series of tests are performed to investigate the evolution of the unconfined compression strength and the Atterberg limits under drying-wetting cycling conditions for specimens treated by iron tailing sands and calcium carbide slag. Typical results of the unconfined compressive strength can be divided into three stages. The unconfined compressive strength increases initially and then decreases to reach a stable state with continuous drying-wetting process. The calcium carbide slag content (αCCS) of 10% can be determined for the minimum effect of the drying-wetting cycle on the strength of the treated specimen. An exponential relationship is established to describe the evolution of the unconfined compressive strength with the drying-wetting cycle. The liquid limit and plastic index of the specimen increase initially followed by a decreasing trend, while a reverse trend was observed for that of the plastic limit during the drying-wetting process. The minimum effect of the drying-wetting cycle on the Atterberg limits can be presented for the specimen with αCCS of 10% as well. [ABSTRACT FROM AUTHOR]

The matching preclusion number of graph G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or almost-perfect matching. The strong matching preclusion number of a graph G is the minimum number of vertices and edges whose deletion leaves the resulting graph without a perfect matching or an almost-perfect matching. The conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or almost-perfect matching. In this paper, we study the matching preclusion number of radix triangular mesh with an odd number of vertices, and strong matching preclusion number and conditional matching preclusion number of radix triangular mesh. Also, we obtained the radix triangular mesh with an even number of vertices is super strongly matched and conditionally super matched. [ABSTRACT FROM AUTHOR]