On Laplacian-energy-like invariant and incidence energy

Document Type: Research Paper

Authors

University of Kashmir

Abstract

For a simple connected graph $G$ with $n$-vertices having Laplacian eigenvalues‎ ‎$\mu_1$‎, ‎$\mu_2$‎, ‎$\dots$‎, ‎$\mu_{n-1}$‎, ‎$\mu_n=0$‎, ‎and signless Laplacian eigenvalues $q_1‎, ‎q_2,\dots‎, ‎q_n$‎, ‎the Laplacian-energy-like invariant($LEL$) and the incidence energy ($IE$) of a graph $G$ are respectively defined as $LEL(G)=\sum_{i=1}^{n-1}\sqrt{\mu_i}$ and $IE(G)=\sum_{i=1}^{n}\sqrt{q_i}$‎. ‎In this paper‎, ‎we obtain some sharp lower and upper bounds for the Laplacian-energy-like invariant and incidence energy of a graph‎.

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F‎. ‎Ayoobi‎, ‎G‎. ‎R‎. ‎Omidi and B‎. ‎Tayfeh-Rezaie (2011). ‎A note on graphs whose signless Laplacian has three distinct eigenvalues. Linear and Multilinear Algebra. 59, 701-706
S‎. ‎B‎. ‎Bozkurt and I‎. ‎Gutman (2013). ‎Estimating the incidence energy. MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem.. 70, 143-156
T‎. ‎A‎. ‎Chishti‎, ‎Hilal A‎. ‎Ganie and S‎. ‎Pirzada (2014). ‎Properties of strong double graphs. J‎. ‎Discrete Math‎. ‎Sci‎. ‎Cryptogr.. 17, 311-319
D‎. ‎Cvetkovic‎, ‎M‎. ‎Doob and H‎. ‎Sachs (1980). ‎Spectra of graphs-Theory and Application. ‎Academic Press‎, ‎New York.
D‎. ‎Cvetkovic‎, ‎P‎. ‎Rowlinson and S‎. ‎K‎. ‎Simic (2007). ‎Signless Laplacians of finite graphs. Linear Algebra Appl.. 423, 155-171
D‎. ‎Cvetkovic and S‎. ‎K‎. ‎Simic (2009). ‎Towards a spectral theory of graphs based on the signless Laplacian‎, ‎I. Publ‎. ‎Inst‎. ‎Math., ‎(Beograd) (N‎. ‎S.). 85, 19-33
S‎. ‎S‎. ‎Dragomir (2003). ‎A survey on Cauchy-Bunyakovsky-Schwarz type discrete inequalities. JIPAM‎. ‎J‎. ‎Inequal‎. ‎Pure Appl‎. ‎Math.. 4, 142
K‎. ‎C‎. ‎Das (2007). ‎A sharp upper bound for the number of spanning trees of a graph. Graphs Combin.. 23, 625-632
M‎. ‎Fiedler (1973). ‎Algebraic Connectivity of Graphs. Czechoslovak Math‎. ‎J.. 23, 298-305
R‎. ‎Grone and R‎. ‎Merris (1994). ‎The Laplacian spectrum of a graph II. SIAM J‎. ‎Discrete Math.. 7, 221-229
I‎. ‎Gutman (2001). ‎The energy of a graph‎: ‎old and new results. ‎Algebraic combinatorics and applications (Gößweinstein‎, ‎1999)‎, ‎Springer‎, ‎Berlin. , 196-211
I‎. ‎Gutman‎, ‎D‎. ‎Kiani and M‎. ‎Mirzakhah (2009). ‎On incidence energy of graphs. MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem.. 62, 573-580
I‎. ‎Gutman‎, ‎D‎. ‎Kiani‎, ‎M‎. ‎Mirzakhah and B‎. ‎Zhou (2009). ‎On incidence energy of a graph. Linear Algebra Appl.. 431, 1223-1233
I‎. ‎Gutman and B‎. ‎Zhou (2006). ‎Laplacian energy of a graph. Linear Algebra Appl.. 414, 29-37
I‎. ‎Gutman‎, ‎B‎. ‎Zhou and B‎. ‎Furtula (2010). ‎The Laplacian-energy like invariant is an energy like invariant. MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem.. 64, 85-96
H‎. ‎A‎. ‎Ganie‎, ‎S‎. ‎Pirzada and I‎. ‎Antal (2014). ‎Energy and Laplacian energy of double graphs. Acta Univ‎. ‎Sap‎. ‎Informatica. 6, 89-116
M‎. ‎Jooyandeh‎, ‎D‎. ‎Kiani and M‎. ‎Mirzakhah (2009). ‎Incidence energy of a graph. MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem.. 62, 561-572
S‎. ‎J‎. ‎Kirkland‎, ‎J‎. ‎J‎. ‎Molitierno‎, ‎M‎. ‎Neumann and B‎. ‎L‎. ‎Shader (2002). ‎On graphs with equal algebraic and vertex connectivity. Linear Algebra Appl.. 341, 45-56
M‎. ‎Liu and B‎. ‎Liu (2012). ‎On sum of powers of the signless Laplacian eigenvalues of graphs. Hacet‎. ‎J‎. ‎Math‎. ‎Stat.. 41, 527-536
J‎. ‎Liu and B‎. ‎Liu (2008). ‎A Laplacian-energy-like invariant of a graph. MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem.. 59, 355-372
M‎. ‎Liu‎, ‎B‎. ‎Liu and X‎. ‎Tan (2014). ‎The first to ninth greatest LEL-invariants of connected graphs. Util‎. ‎Math.. 93, 153-160
A‎. ‎Mohammadian and B‎. ‎Tayfeh-Rezaie (2011). ‎Graphs with four distinct Laplacian eigenvalues. J‎. ‎Algebraic Combin.. 34, 671-682
S‎. ‎Pirzada (2012). ‎An Introduction to Graph Theory. ‎Universities Press‎, ‎Orient Blackswan.
Z‎. ‎Tang and Y‎. ‎Hou (2011). ‎On incidence energy of trees. MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem.. 66, 977-984
W‎. ‎Wang and Y‎. ‎Luo (2012). ‎On Laplacian-energy-like invariant of a graph. Linear Algebra Appl.. 437, 713-721
B‎. ‎Zhou (2010). ‎More upper bounds for the incidence energy. MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem.. 64, 123-128
J‎. ‎Zhang and J‎. ‎Li (2012). ‎New results on the incidence energy of graphs. MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem.. 68, 777-803
B‎. ‎Zhou and A‎. ‎Ilic (2010). ‎On the sum of powers of Laplacian eigenvalues of bipartite graphs. Czechoslovak Math‎. ‎J.. 60, 1161-1169
B‎. ‎X‎. ‎Zhu (2011). ‎The Laplacian-energy like of graphs. Appl‎. ‎Math‎. ‎Lett.. 24, 1604-1607