It is well known that networks are dynamic graphs, and the topology of a network can be described by a graph. Thus, the reliability of a network under edge failure is defined as the probability that its corresponding topological graph remains connected under the condition that the edges fail with independent probabilities. In this paper, the reliability of a class of two-layer networks is considered, where each layer is a complete graph and the edges joining different layers are one-to-one correspondingly connected. The edge failure probability is uniform in the same layer and distinct in different layers and between layers. The recursive formula for the reliability (reliability polynomial) of the two-layer network is obtained, and the corresponding algorithm is also given. Furthermore, the reliability of several networks is computed by Python, which verifies the correctness of the algorithm.

In this paper, we give a method to characterize graphs determined by their adjacency spectrum. At first, we give two parameters ${\Pi }_1\left(G\right)$ and ${\Pi }_2\left(G\right)$, which are related to coefficients of the characteristic polynomial of graph G. All connected graphs with ${\Pi }_1\left(G\right)\in \{1,0,-1,-2,-3\}$ and ${\Pi }_2\left(G\right)\in \{0,-1,-2,-3\}$ are characterized. Some interesting properties of ${\Pi }_1\left(G\right)$ and ${\Pi }_2\left(G\right)$ are also given. We then find the necessary and sufficient conditions for two classes of graphs to be determined by their adjacency spectrum.

A graph with specified target vertices in vertex set is a -terminal graph. The -terminal reliability is the connection probability of the fixed target vertices in a -terminal graph when every edge of this graph survives independently with probability . For the class of two-terminal graphs with a large number of edges, Betrand, Goff, Graves, and Sun constructed a locally most reliable two-terminal graph for close to 1 and illustrated by a counterexample that this locally most reliable graph is not the uniformly most reliable two-terminal graph. At the same time, they also determined that there is a uniformly most reliable two-terminal graph in the class obtained by deleting an edge from the complete graph with two target vertices. This article focuses on the uniformly most reliable three-terminal graph of dense graphs with vertices and edges. First, we give the locally most reliable three-terminal graphs of and in certain ranges for close to 0 and 1. Then, it is proved that there is no uniformly most reliable three-terminal graph with specific and , where and . Finally, some uniformly most reliable graphs are given for vertices and edges, where and or and .

This paper proposes a computational method for the reliability of 2-separable networks. Based on graph theory and probability theory, this method simplifies the calculation process by constructing a network equivalent model and designing corresponding algorithms to achieve the efficient evaluation of reliability. Considering independent random failures of edges with equal probability q, this method can accurately calculate the reliability of 2-separable networks, and its effectiveness and accuracy are verified through examples. In addition, to demonstrate the generality of our method, we have also applied it to other 2-separable networks with fractal structures and proposed linear algorithms for calculating their all-terminal reliability.

A graph $G$ with $k$ specified target vertices in vertex set is a $k$-terminal graph. The $k$-terminal reliability is the connection probability of the fixed $k$ target vertices in a $k$-terminal graph when every edge of this graph survives independently with probability $p$. For the class of two-terminal graphs with a large number of edges, Betrand, Goff, Graves and Sun constructed a locally most reliable two-terminal graph for $p$ close to $1$, and illustrated by a counterexample that this locally most reliable graph is not the uniformly most reliable two-terminal graph. At the same time, they also determined that there is a uniformly most reliable two-terminal graph in the class obtained by deleting an edge from the complete graph with two target vertices. This article focuses on the uniformly most reliable three-terminal graph of dense graphs with $n$ vertices and $m$ edges. First, we give the locally most reliable three-terminal graphs of $n$ and $m$ in certain ranges for $p$ close to $0$ and $1$. Then, it is proved that there is no uniformly most reliable three-terminal graph with specific $n$ and $m$, where $n\geq7$ and $\binom{n}{2}-\lfloor\frac{n-3}{2}\rfloor\leq m\leq\binom{n}{2}-2$. Finally, some uniformly most reliable graphs are given for $n$ vertices and $m$ edges, where $4\leq n\leq 6$ and $m=\binom{n}{2}-2$ or $n\geq5$ and $m=\binom{n}{2}-1$.

A three-terminal graph is defined as a simple graph comprising three specified target vertices. The reliability of three-terminal graphs represents the probability that these three target vertices remain connected, given that each edge fails independently with a constant probability q. In this paper, we focus on exploring the characteristics of more reliable three-terminal graphs when the edge failure probability approaches 1. Three reliability comparison criteria are proposed to characterize the locally most reliable three-terminal graph progressively when the number of edges m is in the range of $[5,4n-10]$ and $[\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)-n+4,\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)]$. At the same time, the locally optimal structures in the range of the edge number m with $(4n-10,\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)-n+4)$ are restricted to six specific classes of graphs. Furthermore, based on these criteria, a method is introduced to search local optimal structures and offer a theoretical foundation for constructing optimal networks and repairing faulty ones.