Adjacent1

Adjacent1 is a coined term used in mathematical and computational contexts to describe a neighboring element in a sequence or structure when indexing begins at one. It is not a standard formal concept, but serves as a convenient label for discussions of immediate successor relations in one-based environments.

In formal terms, consider a finite sequence or array indexed from 1 to n, written as a1, a2, ..., an. The adjacent1 relation identifies, for a given position i, the element at the next position, provided that i is less than n. Thus adjacent1(ai) yields ai+1 when it exists, and is undefined or null for the last element an. This usage mirrors the idea of “next element” in one-based indexing, distinguishing it from similar notions in zero-based systems.

In graph theory and data structures, adjacent1 can describe the neighbor relationship in structures that use one-based indexing for node identifiers or positions. For example, in a linear chain of nodes labeled 1 through n, the adjacent1 relation connects i to i+1 for i in 1 to n−1. In grids or matrices that adopt one-based indexing, adjacent1 may be used to articulate moves or transitions to the element immediately to the right or forward along a single dimension, depending on the context.

Applications of adjacent1 appear in algorithm descriptions, such as traversals, neighbor searches, and iterative procedures that rely on stepping to the next position. Because adjacent1 is not a universal standard, its precise definition should be stated within each formal work to avoid ambiguity.

See also: adjacency, neighbor, one-based indexing.