Sinha Deepa, Somra Sachin
Department of Mathematics, Faculty of Mathematics and Computer Science, South Asian University, New Delhi 110068, India.
MethodsX. 2025 Jan 14;14:103160. doi: 10.1016/j.mex.2025.103160. eCollection 2025 Jun.
A signed graph is a pair that consists of a graph and a sign mapping called signature from to the sign group . In this paper, we discuss the -path product signed graph where vertex set of is the same as that of and two vertices are adjacent if there is a path of length , between them in the signed graph . The sign of an edge in the -path product signed graph is determined by the product of marks of the vertices in the signed graph , where the mark of a vertex is the product of signs of all edges incident to it. In this paper, we provide a characterization of which are switching equivalent to -path product signed graphs for which are switching equivalent to and also the negation of the signed graph ŋ that are switching equivalent to for . We also characterize signed graphs that are switching equivalent to -distance signed graph for where 2-distance signed graph defined as follows: the vertex set is same as the original signed graph and two vertices , are adjacent if and only if there exists a distance of length two in . The edge is negative if and only if all the edges, in all the distances of length two in are negative otherwise the edge is positive. The -path network along with these characterizations can be used to develop model for the study of various real life problems communication networks.•-path product signed graph.•-distance signed graph.
带符号图是一个由图(G)和一个称为签名的符号映射(\sigma)组成的对，该映射从(G)的边集(E(G))到符号群({ +, - })。在本文中，我们讨论(k -)路径乘积带符号图(G^k_{\sigma})，其顶点集与(G)相同，并且如果在带符号图(G)中它们之间存在长度为(k)的路径，则两个顶点相邻。(k -)路径乘积带符号图中一条边的符号由带符号图(G)中顶点的标记的乘积确定，其中一个顶点的标记是与它相关联的所有边的符号的乘积。在本文中，我们给出了与(k -)路径乘积带符号图(G^k_{\sigma})切换等价的图(G)的特征，其中(G)与(H)切换等价，并且还给出了与(H)的带符号图(\overline{H})的否定切换等价的图(G)的特征。我们还刻画了与(k -)距离带符号图(G^k_d)切换等价的带符号图，其中(2 -)距离带符号图(G^2_d)定义如下：顶点集与原始带符号图(G)相同，并且当且仅当在(G)中存在长度为二的距离时，两个顶点(u)，(v)相邻。边((u, v))为负当且仅当在(G)中所有长度为二的距离中的所有边均为负，否则该边为正。(k -)路径网络以及这些特征可用于开发用于研究各种现实生活问题如通信网络的模型。
• (k -)路径乘积带符号图。
• (k -)距离带符号图。
