Construction of larger singular and nonsingular graphs using a path
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https://doi.org/10.26637/MJM0804/0049Abstract
A singular graph \(G\) has an adjacency matrix \(A(G)\) with nullity \(\eta(G)>0\). Vertices of singular graphs are classified as core and noncore vertices. There are two types of noncore vertices: noncore vertices of zero null spread and of null spread -1 . Deletion of these vertices from a singular graph either changes the nullity or leave it unaltered. In this paper larger singular and nonsingular graphs were constructed by joining singular graphs by a path. As singular graphs have different types of vertices, the graphs constructed in this way differ in nullity depending on the vertex we are joining during construction. An attempt was made to construct singular graph of maximum nullity. Various spectral properties of the resulting graphs were studied.
Keywords:
Singular graph, Path, Nullity, Core vertices, CoalescenceMathematics Subject Classification:
Mathematics- Pages: 1614-1621
- Date Published: 01-10-2020
- Vol. 8 No. 04 (2020): Malaya Journal of Matematik (MJM)
S. Akbari, E. Ghorbani, S. Zare, Some relation between rank, chromatic number and energy of graphs, Discrete Mathematics, 309 (3)(2009), 601-605.
D. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs - Theory and Application, Academic Press, New York, (1980).
C. J. Edholm, L. Hogben, M. Huynh, J. LaGrande, D. D. Row, Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph, Preprint, (2010).
C. Godsil, G. Royle, Algebraic Graph Theory, SpringerVerlag, New York, 2001.
X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, (2012).
T. K. Mathew Varkey, John, K. Rajan, On the spectrum and energy of coalesced singular graphs, Bulletin of Kerala Mathematics Association, 13 (1) (2016), p37-50.
T. K. Mathew Varkey, John, K. Rajan, On the Spectrum and Energy of Singular Graphs, AKCE International Journal of Graphs and Combinatorics, 16 (2019), 265-271.
T. K. Mathew Varkey, John, K. Rajan, On the Construction of Larger Singular Graphs, Global Journal of Pure and Applied Mathematics, 13(7)(2017), 2875-2891.
A. J. Schwenk, Computing the Characteristic Polynomial of a Graph, in: Bari, R.A., Harary, F., eds., Graphs and Combinatorics, Springer-Verlag, pp. 153-172, 1974.
A. J. Schwenk, R. J. Wilson, On the Eigen Values of a Graph, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory, Academic Press, London, pp. 307-336, 1978.
I. Sciriha, On the construction of graphs of nullity one, Discrete Mathematics, 181(1988), 193-211.
I. Sciriha, I. Gutman, Minimal configurations and interlacing, Graph Theory Notes of New York XLIV, (2005), $38-40$.
I. Sciriha, A characterization of singular graphs, Electronic Journal of Linear Algebra, 16(2007), 451-462.
I. Sciriha, Coalesced and embedded nut graphs in singular graphs, ARS Mathematica Contemporary, 1(2008), 20-31.
I. Sciriha, Extremal non-bonding orbitals, MATCH Commun. Math. Comput. Chem., 63(2009), 751-768.
I. Triantafillou, On the energy of singular graphs, Electric Journal of Linear Algebra, 26(2013), 535-545.
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