Within computing and information technology, interconnection systems are often modeled using graphs. A notable example is the swapped Optical Transpose Interconnection System (OTIS). Ensuring resilience to faults is crucial in optoelectronic systems. Across different fault classifications that can arise in an interconnection structure, two notable ones are the failure of a node (such as a processor in the case of OTIS over base graph G, denoted by OG and the disruption in facilitating communication between nodes breakdown of processor-to-processor communication). To manage these faults, it is crucial to assign a unique identifier to each node. In terms of graph theory, this corresponds to identifying the metric dimension, denoted as beta(G), and Fault-oriented metric dimension beta '(G) of the graph G representing the interlinking structure. This paper analyzes the OTIS architecture based on Cm (Cycle graph containing m vertices), represented as OCm, for resolvability and resolvability under fault conditions. It is proved that beta(OCm)=m-1 and beta '(OCm)=m.
A quasi total double Roman dominating function (QTDRD-function) on a graph G=(V(G),E(G)) is a function f:V(G)->{0,1,2,3} having the property that (i) if f(v)=0, then the vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f(w)=3, and if f(v)=1, then the vertex v must have at least one neighbor w with f(w)>= 2, and (ii) if x is an isolated vertex in the subgraph of G induced by the set of vertices assigned nonzero values, then f(x)=2. The weight of a QTDRD-function f is the sum of its function values over the whole vertices, and the quasi total double Roman domination number gamma qtdR(G) equals the minimum weight of a QTDRD-function on G. The gamma qtdR-stability st gamma qtdR(G) (resp. gamma qtdR--stability st gamma qtdR-(G), gamma qtdR+-stability st gamma qtdR+(G)) of G is defined as the order of the smallest set of vertices whose removal changes (resp. decreases, increases) the quasi total double Roman domination number. In this paper, we first determine the exact values of gamma qtdR-stability for some special classes of graphs, and then we present some bounds on st gamma qtdR(G) and st gamma qtdR-(G). We also characterize the graphs with large st gamma qtdR-(G). Moreover, we show that if T is a nontrivial tree, then st gamma qtdR(T)<= 2, and if further T has maximum degree Delta >= 3, then st gamma qtdR-(T)<=Delta-1.
This paper investigates methods for constructing graphs whose matching polynomials are irreducible over Q. Building on this, the construction method is extended to general graph polynomials, for graphs whose polynomials satisfy certain properties, the connection construction produces a series of irreducible polynomial, such as characteristic polynomials and matching polynomials. Finally, the paper presents algorithms for calculating the matching polynomial of a given graph and the matching polynomial for a given number of vertices, implemented in Python.
Let T be a graph obtained by taking the Cartesian product of n cycles of even lengths >= 4. It is known that for any subset of k vertices of T with 1 <= k <= 2n-1, there exists a 2-factor in T where each cycle contains exactly one of the k vertices. We generalize this result to r-factors by proving that for 1 <= k <= 2n-r+1 and 2 <= r <= 2n, there exists an r-factor H in T such that each component of H is an r-regular, r-connected and bipancyclic graph, with each component containing exactly one of the k specified vertices of T. As a consequence, we obtain an analogous result for hypercubes.
In order to study the fault-tolerant capability of the multiprocessor system, several new self-diagnosis fault-tolerant models are proposed. In particular, non-inclusive diagnosability was showed by Ding et.al. in 2020. The non-inclusive diagnosability of regular graphs with C-3-free was established by Xu et. al. in 2022. In this paper, the non-inclusive diagnosability t(N)(AN(n)) of alternating group network is determined under the PMC model and the MM* model.
