Spectral extrema of graphs with fixed size: Cycles and complete bipartite graphs

Abstract Nikiforov (2002) showed that if G is K r + 1 -free then the spectral radius ρ ( G ) ≤ 2 m ( 1 − 1 ∕ r ) , which implies that G contains C 3 if ρ ( G ) > m . In this paper, we follow this direction in determining which subgraphs will be contained in G if ρ ( G ) > f ( m ) , where f ( m ) ∼ m as m → ∞ . We first show that if ρ ( G ) ≥ m , then G contains K 2 , r + 1 unless G is a star; and G contains either C 3 + or C 4 + unless G is a complete bipartite graph, where C t + denotes the graph obtained from C t and C 3 by identifying an edge. Secondly, we prove that if ρ ( G ) ≥ 1 2 + m − 3 4 , then G contains pentagon and hexagon unless G is a book; and if ρ ( G ) > 1 2 ( k − 1 2 ) + m + 1 4 ( k − 1 2 ) 2 , then G contains C t for every t ≤ 2 k + 2 . In the end, some related conjectures are provided for further research.

[1]  P. Erdos,et al.  On maximal paths and circuits of graphs , 1959 .

[2]  Xin Li,et al.  Sharp upper bounds of the spectral radius of a graph , 2019, Discret. Math..

[3]  Vladimir Nikiforov Walks and the spectral radius of graphs , 2005 .

[4]  László Babai,et al.  Spectral Extrema for Graphs: The Zarankiewicz Problem , 2009, Electron. J. Comb..

[5]  M. Simonovits,et al.  The History of Degenerate (Bipartite) Extremal Graph Problems , 2013, 1306.5167.

[6]  Eigenvalues and triangles in graphs , 2019, Combinatorics, Probability and Computing.

[7]  Herbert S. Wilf,et al.  Spectral bounds for the clique and independence numbers of graphs , 1986, J. Comb. Theory, Ser. B.

[8]  A. Hoffman,et al.  On the spectral radius of (0,1)-matrices , 1985 .

[9]  Xing Peng,et al.  Extensions of the Erdős–Gallai theorem and Luo’s theorem , 2019, Combinatorics, Probability and Computing.

[10]  Shaowei Sun,et al.  A conjecture on the spectral radius of graphs , 2020 .

[11]  Vladimir Nikiforov,et al.  Some Inequalities for the Largest Eigenvalue of a Graph , 2002, Combinatorics, Probability and Computing.

[12]  V. Nikiforov The maximum spectral radius of C4-free graphs of given order and size , 2007, 0712.1301.

[13]  Michael Tait,et al.  Three conjectures in extremal spectral graph theory , 2016, J. Comb. Theory, Ser. B.

[14]  Richard A. Brualdi,et al.  On the Spectral Radius of (0, 1)-Matrices with 1's in Prescribed Positions , 1996, SIAM J. Matrix Anal. Appl..

[15]  Michael Tait,et al.  Degenerate Turán Problems for Hereditary Properties , 2017, Electron. J. Comb..

[16]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[17]  V. Nikiforov Some new results in extremal graph theory , 2011, 1107.1121.

[18]  Britta Papendieck,et al.  On maximal entries in the principal eigenvector of graphs , 2000 .

[19]  Vladimir Nikiforov,et al.  A spectral condition for odd cycles in graphs , 2007, 0707.4499.

[20]  Peter Rowlinson On the maximal index of graphs with a prescribed number of edges , 1988 .

[21]  Eva Nosal,et al.  Eigenvalues of graphs , 1970 .

[22]  Vladimir Nikiforov,et al.  The spectral radius of graphs without paths and cycles of specified length , 2009, 0903.5351.