WU Yaping1, GUO Huiyi2, YUAN Shuai 1. School of Mathematics and Computer Science, Jianghan University, Wuhan 430056, Hubei, China; 2. Department of Mathematics, University of Washington, Seattle, WA 98195, USA; 3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA

Abstract:

A connected graph G = (V(G), E(G)) is called a quasi-unicyclic graph, if there exists u0 ∈ V(G) such that G − u0 is a unicyclic graph. Denote Q(n, d0) = {G: G is a quasi-unicyclic graph of order n with G − u0 being a unicyclic graph and dG(u0) = d0}. Let A(G) be the adjacency matrix of a graph G, and let λ1(G), λ2(G),…, λn(G) be the eigenvalues in non-increasing order of A(G). The number (sumlimits_{i = 1}^n {lambda _i^k(G)} ) (k = 0,1, …, n−1) is called the k-th spectral moment of G, denoted by Sk (G). Let S (G) = (S0(G), S1(G),…, Sn−1(G)) be the sequence of spectral moments of G. For two graphs G1, G2, we have G1 ≺SG2 if for some k(k = 1,2,…, n−1), and we have Si(G1) = Si(G2) (i = 0,1, …, k−1) and Sk(G1) < Sk(G2). In this paper, we determine the second to the fourth largest quasi-unicyclic graphs, in an S-order, in the set Q(n, d0), respectively.

[1] Bondy J A, Murty U S R. Graph Theory with Applications[M]. London: Macmillan, 1976.
[2] Cvetkovi´c D, Doob M, Sachs H. Spectra of Graphs—Theory and Applications [M]. New York: Academic Press, 1980.
[3] Harary F. Graph Theory [M]. Reading, MA: Addison- Wes-ley, 1968.
[4] Gutman I, Rosenfeld V R. Spectral moments of polymer graphs [J]. Theoretica Chemica Acta, 1996, 93: 191-197.
[5] Gutman I, Trinajsti´c N. Graph theory and molecular. Total electron energy of alternant hydrocarbons [J]. Chemical Physics Letters, 1972, 17: 535-538.
[6] Cvetkovi´c D, Doob M, Sachs H, et al. Recent Results in the Theory of Graph Spectra, Annals of Discrete Mathematics Series [M]. Amsterdam: North-Holland, 1988.
[7] Cvetkovi´c D, Petri´c M. A table of connected graphs on six vertices [J]. Discrete Mathematics, 1984, 50: 37-49.
[8] Cvetkovi´c D, Rowlinson P. Spectra of unicyclic graphs [J]. Graph and Combinatorics, 1987, 7: 7-23.
[9] Fan Q, Wu Y P. Spectral moments sequence and lexico-graphical order of graphs [J]. Journal of Wuhan University (Nat Sci Ed), 2009, 55(6): 625-628(Ch).
[10] Wu Y P, Fan Q. On the lexicographical ordering by spectral moments of bicyclic graphs [J]. Ars Combinatoria, 2014, 114: 213-222.
[11] Wu Y P, Liu H Q. Lexicographical ordering by spectral moments of trees with a prescribed diameter [J]. Linear Al-gebra and Its Applications, 2010, 433: 1707-1713.
[12] Pan X F, Hu X L, Liu X G, et al. The spectral moments of trees with given maximum degree [J]. Applied Mathematics Letters, 2011, 24: 1265-1268.
[13] Cheng B, Liu B L. Lexicographical ordering by spectral moments of trees with k pendant vertices and integer partitions [J]. Applied Mathematics Letters, 2012, 25: 858-861.
[14] Cheng B, Liu B L, Liu J X. On the spectral moments of uni-cyclic graphs with ﬁxed diameter [J]. Linear Algebra and Its Applications, 2012, 437: 1123-1131.
[15] Andriantiana E, Wagner S. Spectral moments of trees with given degree sequence [J]. Linear Algebra and Its Applications, 2013, 439: 3980-4002.
[16] Li S C, Song Y B. On the spectral moment of trees with given degree sequences [EB/OL]. [2012-09-11]. http:// arxiv.org/ abs/1209.2188.
[17] Hu S N, Li S C. On the spectral moment of graphs with given clique number [J]. Rocky Mountain Journal of Mathematics, 2016, 46: 216-282.
[18] Li S C, Zhang J J. Lexicographical ordering by spectral moments of trees with a given bipartition [J]. Bulletin Iranian Mathematical Society, 2014, 40: 1027-1045.
[19] Li S C, Zhang H H, Zhang M J. On the spectral moment of graphs with k cut edges [J]. Electronic Journal of Linear Al-gebra, 2013, 26: 718-731.
[20] Pan X F, Liu X G, Liu H Q. On the spectral moment of quasi- trees [J]. Linear Algebra and Its Applications, 2012, 436: 927-934.
[21] Wu Y P, Fan Q, Yuan S. On the spectral moment of quasi-trees and quasi-unicyclic graphs [J]. Ars Combinatoria, 2018, 137: 335-344.
[22] Nie Z B. Ordering Graphs with Respect to Spectral Moments [D]. Changsha: Normal University of Hunan, 2008(Ch).