Writer identification is an active research problem due to its applications in forensic and historic documents analysis. It is challenging to identify a writer from her handwritten characters' shapes produced via practiced writing style. Different writing shapes, styles, orientations, various sizes of characters, complex structures, inconsistency, and cursive nature of the text make it a tougher undertaking. To solve this problem, we need to explore a structural representation and spatial information of the handwritten characters. For this, a novel graph-based approach is proposed here to spatially map the handwritten text, adapt its structure, size, and explore the relationship that exist between them. First, image processing steps such as binarization, baseline correction, separation of the writing region, and thinning of the strokes to a width of a single pixel are executed. This work presents a novel algorithm for detecting key points (KPs) in a handwritten skeleton image and extracting their two-dimensional pixel coordinates values. The handwriting samples are then transformed into a graph-based representation with KPs representing nodes and the line segments connecting adjacent KPs as the edges. Features are extracted from the graph-based representations of the handwritten text. For classification, ensemble learning approaches are employed. Four benchmark datasets and one custom collected dataset are utilized for experimentations. The proposed solution achieves identification accuracies of 98.26%, 98.84%, 99.67%, 98.51%, and 97.73%, on CERUG-EN, CVL, Firemaker, IAM, and custom datasets, respectively.