**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**32912

##### Bounds on the Second Stage Spectral Radius of Graphs

**Authors:**
S.K.Ayyaswamy,
S.Balachandran,
K.Kannan

**Abstract:**

Let G be a graph of order n. The second stage adjacency matrix of G is the symmetric n × n matrix for which the ijth entry is 1 if the vertices vi and vj are of distance two; otherwise 0. The sum of the absolute values of this second stage adjacency matrix is called the second stage energy of G. In this paper we investigate a few properties and determine some upper bounds for the largest eigenvalue.

**Keywords:**
Second stage spectral radius,
Irreducible matrix,
Derived graph

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1071578

**References:**

[1] R. Balakrishnan, The energy of graph, Linear Algebra Appl. 387(2004) 287-295.

[2] D.Cvetkovic, M. Doob, H. Saches, Spectra of Graphs- Theory and Application, third ed., Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995.

[3] Dasong Cao, Bounds on Eigenvalues and Chromatic Numbers, Linear Algebra and its Applications, 270 (1998), 1-13.

[4] M.N.Ellingham and X.Zha, The spectral radius of graphs on surfaces, J.Combin.Theory Series B 78(2000), 45-56.

[5] D. Stevanovic, The largest eigenvalue of nonregular graphs, J.Combin.Theory B 91(2004) 143-146.

[6] Yuan Hong and Jin-Long Shu, A Sharp Upper Bound of the Spectral Radius of Graphs, Journal of Combinatorial Theory, Series B 81,177- 183(2001).