A simple connected non-regular graph is said to be (Formula presented.) -bidegreed, or biregular, if the vertices have degree from the set (Formula presented.) , with (Formula presented.). We consider two classes of (Formula presented.) -bidegreed graphs denoted by (Formula presented.) and (Formula presented.). A graph belongs to (Formula presented.) if: a) it is obtained from (Formula presented.) disjoint paths (Formula presented.) , (Formula presented.) , by identifying the vertices (Formula presented.) and the vertices (Formula presented.) , and the graph so obtained has n vertices; b) for each of the n vertices, pendant vertices are added, so that any vertex from any (Formula presented.) has degree (Formula presented.). The class (Formula presented.) , (Formula presented.) , is similarly obtained by identifying all the vertices (Formula presented.) and (Formula presented.) from the (Formula presented.) ’s, into a single vertex. In this paper, we show that for any graph in (Formula presented.) or (Formula presented.) , the spectral radius of the adjacency matrix increases whenever the difference between the lengths of any two (Formula presented.) ’s increases. We also compute some bounds for the spectral radius when the lengths of the (Formula presented.) ’s tend to infinity. Finally, we discuss about bicyclic (Formula presented.) -bidegreed graphs with n degree (Formula presented.) vertices minimizing the spectral radius. We prove that in most cases such graphs do not belong to (Formula presented.).

On the largest eigenvalue of some bidegreed graphs / Belardo, Francesco. - In: LINEAR & MULTILINEAR ALGEBRA. - ISSN 0308-1087. - 63:1(2015), pp. 166-184. [10.1080/03081087.2013.860591]

On the largest eigenvalue of some bidegreed graphs

BELARDO, Francesco
2015

Abstract

A simple connected non-regular graph is said to be (Formula presented.) -bidegreed, or biregular, if the vertices have degree from the set (Formula presented.) , with (Formula presented.). We consider two classes of (Formula presented.) -bidegreed graphs denoted by (Formula presented.) and (Formula presented.). A graph belongs to (Formula presented.) if: a) it is obtained from (Formula presented.) disjoint paths (Formula presented.) , (Formula presented.) , by identifying the vertices (Formula presented.) and the vertices (Formula presented.) , and the graph so obtained has n vertices; b) for each of the n vertices, pendant vertices are added, so that any vertex from any (Formula presented.) has degree (Formula presented.). The class (Formula presented.) , (Formula presented.) , is similarly obtained by identifying all the vertices (Formula presented.) and (Formula presented.) from the (Formula presented.) ’s, into a single vertex. In this paper, we show that for any graph in (Formula presented.) or (Formula presented.) , the spectral radius of the adjacency matrix increases whenever the difference between the lengths of any two (Formula presented.) ’s increases. We also compute some bounds for the spectral radius when the lengths of the (Formula presented.) ’s tend to infinity. Finally, we discuss about bicyclic (Formula presented.) -bidegreed graphs with n degree (Formula presented.) vertices minimizing the spectral radius. We prove that in most cases such graphs do not belong to (Formula presented.).
2015
On the largest eigenvalue of some bidegreed graphs / Belardo, Francesco. - In: LINEAR & MULTILINEAR ALGEBRA. - ISSN 0308-1087. - 63:1(2015), pp. 166-184. [10.1080/03081087.2013.860591]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11588/618957
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