2022/02/28 by V. Vilfred Kamalappan, Lowell W. Beineke, Kamalappan, V. Vilfred +5
Computer Science · Neuroscience · #Graph Labeling and Dimension Problems #Nuclear Receptors and Signaling
paper · pdf · doi:10.48550/arxiv.2203.00409
Frank Harary introduced the concept of integral sum graph. A graph G is an integral sum graph if its vertices can be labeled with distinct integers so that e = uv is an edge of G if and only if the sum of the labels on vertices u and v is also a label in G. For any non-empty set of integers S, let G+(S) denote the integral sum graph on the set S. In G+(S), we define an edge-sum class as the set of all edges each with same edge sum number and call G+(S) an edge sum color graph if each edge-sum class is considered as an edge color class of G+(S). The number of distinct edge-sum classes of G+(S) is called its edge sum chromatic number. The main results of this paper are (i) the set of all edge-sum classes of an integral sum graph partitions its edge set; (ii) the edge chromatic number and the edge sum chromatic number are equal for the integral sum graphs G0,s and Sn, Star graph of order n, whereas it is not in the case of Gr,s = G+([r,s]), r < 0 < s, n,s ≥ 2, n,r,s∈ℕ. We also obtain an interesting integral sum labeling of Star graphs.