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The graphs with exactly two distance eigenvalues different from \(-1\) and \(-3\)

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Abstract

In this paper, we completely characterize the graphs with third largest distance eigenvalue at most \(-1\) and smallest distance eigenvalue at least \(-3\). In particular, we determine all graphs whose distance matrices have exactly two eigenvalues (counting multiplicity) different from \(-1\) and \(-3\). It turns out that such graphs consist of three infinite classes, and all of them are determined by their distance spectra. We also show that the friendship graph is determined by its distance spectrum.

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Acknowledgments

The authors are grateful to the referees for their helpful comments and suggestions that have improved our results in this paper.

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Authors and Affiliations

  1. College of Mathematics and Systems Science, Xinjiang University, Ürümqi, 830046, Xinjiang, People’s Republic of China

    Lu Lu, Qiongxiang Huang & Xueyi Huang

Authors
  1. Lu Lu
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  2. Qiongxiang Huang
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  3. Xueyi Huang
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Corresponding author

Correspondence to Qiongxiang Huang.

Additional information

Supported by the National Natural Science Foundation of China (11671344, 11261059, 11531011).

Appendix: Spectra of graphs in \({\mathcal {G}}[-3\le \partial _n,\partial _3\le -1]\)

Appendix: Spectra of graphs in \({\mathcal {G}}[-3\le \partial _n,\partial _3\le -1]\)

Graphs

Distance Spectra

f(x)

\(S(0,1)=K_1\vee (K_5\cup K_1)\)

\([7.66,-0.71,(-1)^4,-2.96]\)

–

\(S(1,0)=K_2\vee (K_5\cup K_1)\)

\([8.47,-0.47,(-1)^5,-3]\)

–

\(S(m,n)=(K_5\cup K_1)\vee (mK_2\cup nK_1)\,(m,n\ne 0)\)

\([\partial _1,\partial _2,(-1)^{m+4},(-2)^{n-1},\partial _3,(-3)^m]\)

[\(x^3-(2n+4m+2)x^2+(2n+8m-28)x+(32m+24n-40)\)]

\(S(m,0)=(K_5\cup K_1)\vee mK_2\,(m\ge 2)\)

\([\partial _1,\partial _2,(-1)^{m+4},(-3)^m]\)

[\(-x^2+(4m+4)x+20-16m\)]

\(S(0,n)=(K_5\cup K_1)\vee nK_1\,(n\ge 2)\)

\([\partial _1,\partial _2,(-1)^4,\partial _3,(-2)^{n-1}]\)

[\(x^3-(2n+2)x^2+(2n-28)x+(24n-40)\)]

\(K_r\vee T_1=K_r\vee (K_4\cup K_1)\)

\([\partial _1,\partial _2,(-1)^{r+2},\partial _3](\partial _3>-3)\)

[\(x^3-(r+2)x^2-(2r+19)x+(3r-16)\)]

\(K_r\vee T_2=K_r\vee (K_3\cup 2K_1)\)

\([\partial _1,\partial _2,(-1)^{r+1},-2,\partial _3](\partial _3>-3)\)

[\(x^3-(r+3)x^2-(r+24)x+(6r-20)\)]

\(K_r\vee T_3=K_r\vee (K_3\cup K_1)\)

\([\partial _1,\partial _2,(-1)^{r+1},\partial _3](\partial _3>-3)\)

[\(x^3-(r+1)x^2-(2r+14)x+(2r-12)\)]

\(K_r\vee T_4(m,n)=K_r\vee (mK_2\cup nK_1)\,(n,m\ne 0)\)

\([\partial _1,\partial _2,(-1)^{m+r-1},(-2)^{n-1},\partial _3,(-3)^{m-1}]\)

[\(x^3+(6-2n-4m-r)x^2+(2mr-8n-5r-12m+nr+11)x-(8m+6n+6r-4mr-3nr-6)\)]

\(K_r\vee T_4(m,0)=K_r\vee mK_2\,(m\ge 2)\)

\([\partial _1,\partial _2,(-1)^{m+r-1},(-3)^{m-1}]\)

[\(x^2+(4-r-4m)x+(2mr-4m-3r+3)\)]

\(K_r\vee T_4(0,n)=K_r\vee nK_1\,(n\ge 2)\)

\([\partial _1,\partial _2,(-1)^{r-1},(-2)^{n-1}]\)

[\(x^2+(3-r-2n)x+(nr-2n-2r+2)\)]

\(T_1\vee T_1=(K_4\cup K_1)\vee (K_4\cup K_1)\)

\([10.71,1,(-1)^6,-2.71,-3]\)

–

\(T_1\vee T_2=(K_4\cup K_1)\vee (K_3\cup 2K_1)\)

\([11.32,1.46,(-1)^5,-2,-2.78,-3]\)

–

\(T_1\vee T_3=(K_4\cup K_1)\vee (K_3\cup K_1)\)

\([9.65,0.85,(-1)^5,-2.6,-2.9]\)

–

\(T_1\vee T_4(m,n)=(K_4\cup K_1)\vee (mK_2\cup nK_1)\,(m,n\ne 0)\)

\([\partial _1,\partial _2,(-1)^{m+3},(-2)^{n-1},\partial _3,\partial _4,(-3)^{m-1}]\)

[\(x^4+(2-2n-4m)x^3-(6m+5n+25)x^2+(42m+22n-98)x+76m+57n-96\)]

\(T_1\vee T_4(m,0)=(K_4\cup K_1)\vee mK_2\,(m\ge 2)\)

\([\partial _1,\partial _2,(-1)^{m+3},\partial _3,(-3)^{m-1}]\)

[\(x^3-4mx^2+(2m-25)x+38m-48\)]

\(T_1\vee T_4(0,n)=(K_4\cup K_1)\vee nK_1\,(n\ge 2)\)

\([\partial _1,\partial _2,(-1)^3,(-2)^{n-1},\partial _3](\partial _3>-3)\)

[\(x^3-(2n+1)x^2+(n-22)x+19n-32\)]

\(T_2\vee T_2=(K_3\cup 2K_1)\vee (K_3\cup 2K_1)\)

\([11.87,2,(-1)^4,(-2)^2,-2.87,-3]\)

–

\(T_2\vee T_3=(K_3\cup 2K_1)\vee (K_3\cup K_1)\)

\([10.34,1.25,(-1)^4,-2,-2.63,-2.95]\)

–

\(T_2\vee T_4(m,n)=(K_3\cup 2K_1)\vee (mK_2\cup nK_1)\,(m,n\ne 0\))

\([\partial _1,\partial _2,(-1)^{m+2},(-2)^n,\partial _3,\partial _4,(-3)^{m-1}]\)

[\(x^4+(1-2n-4m)x^3-(2m+3n+34)x^2+(64m+35n-124)x+104m+78n-120\)]

\(T_2\vee T_4(m,0)=(K_3\cup 2K_1)\vee mK_2\,(m\ge 2)\)

\([\partial _1,\partial _2,(-1)^{m+2},-2,\partial _3,(-3)^{m-1}]\)

[\(x^3-(4m+1)x^2+(6m-32)x+52m-60\)]

\(T_2\vee T_4(0,n)=(K_3\cup 2K_1)\vee nK_1\,(n\ge 2)\)

\([\partial _1,\partial _2,(-1)^2,(-2)^n,\partial _3](\partial _3>-3)\)

[\(x^3-(2n+2)x^2+(3n-28)x+26n-40\)]

\(T_3\vee T_3=(K_3\cup K_1)\vee (K_3\cup K_1)\)

\([8.57,0.73,(-1)^4,-2.57,-2.73]\)

–

\(T_3\vee T_4(m,n)=(K_3\cup K_1)\vee (mK_2\cup nK_1)\,(m,n\ne 0)\)

\([\partial _1,\partial _2,(-1)^{m+2},(-2)^{n-1},\partial _3,\partial _4,(-3)^{m-1}]\)

[\(x^4+(3-2n-4m)x^3-(8m+6n+16)x^2+(28m+14n-72)x+56m+42n-72\)]

\(T_3\vee T_4(m,0)=(K_3\cup K_1)\vee mK_2\,(m\ge 2)\)

\([\partial _1,\partial _2,(-1)^{m+2},\partial _3,(-3)^{m-1}]\)

[\(x^3-(4m-1)x^2-18x+28m-36\)]

\(T_3\vee T_4(0,n)=(K_3\cup K_1)\vee nK_1\,(n\ge 2)\)

\([\partial _1,\partial _2,(-1)^2,(-2)^{n-1},\partial _3](\partial _3>-3)\)

[\(x^3-2nx^2-16x+14n-24\)]

\(T_4(m_1,n_1)\vee T_4(m_2,n_2)=(m_1K_2\cup n_1K_1)\vee (m_2K_2\cup n_2K_1)(m_1,m_2,n_1,n_2\ne 0\))

\([\partial _1,\partial _2,(-1)^{m_1+m_2},(-2)^{n_1+n_2-2},\partial _3,\partial _4,(-3)^{m_1+m_2-2}]\)

[\(x^4+(10-4(m_1+m_2)-2(n_1+n_2))x^3+(12m_1m_2-28(m_1+m_2)-16(n_1+n_2) +6(m_1n_2+m_2n_1)+37)x^2+(48m_1m_2-64(m_1+m_2)-42(n_1+n_2)+30(m_1n_2+m_2n_1)+18n_1n_2+60)x -(48(m_1+m_2-m_1m_2)+36(n_1+n_2)-36(m_1n_2+m_2n_1)-27n_1n_2+36)\)]

\(T_4(m_1,n_1)\vee T_4(m_2,0)=(m_1K_2\cup n_1K_1)\vee m_2K_2 (m_1,n_1\ne 0,m_2\ge 2)\)

\([\partial _1,\partial _2,(-1)^{m_1+m_2},(-2)^{n_1-1},\partial _3,(-3)^{m_1+m_2-2}]\)

[\(x^3+(8-4(m_1+m_2)-2n_1)x^2+(12m_1m_2-20(m_1+m_2)-12n_1+6m_2n_1+21)x -24(m_1+m_2)+18m_2n_1+24m_1m_2-18n_1+18\)]

\(T_4(m_1,n_1)\vee T_4(0,n_2)=(m_1K_2\cup n_1K_1)\vee n_2K_1 (m_1,n_1\ne 0,n_2\ge 2)\)

\([\partial _1,\partial _2,(-1)^{m_1},(-2)^{n_1+n_2-2},\partial _3,(-3)^{m_1-1}]\)

[\(x^3+(7-2(n_1+n_2)-4m_1)x^2+(6m_1n_2-10(n_1+n_2)-16m_1+3n_1n_2+16)x-12(n_1+n_2)+12m_1n_2+9n_1n_2-16m_1+12\)]

\(T_4(m_1,0)\vee T_4(m_2,0)=m_1K_2\vee m_2K_2\,(m_1,m_2\ge 2)\)

\([\partial _1,\partial _2,(-1)^{m_1+m_2},(-3)^{m_1+m_2-2}]\)

\([x^2+(6-4(m_1+m_2))x-12(m_1+m_2)+12m_1m_2+9]\)

\(T_4(m_1,0)\vee T_4(0,n_2)=m_1K_2\vee n_2K_1\,(m_1,n_2\ge 2)\)

\([\partial _1,\partial _2,(-1)^{m_1},(-2)^{n_2-1},(-3)^{m_1-1}]\)

\([x^2+(6-4(m_1+m_2))x-12(m_1+m_2)+12m_1m_2+9]\)

\(T_4(0,n_1)\vee T_4(0,n_2)=n_1K_2\vee n_2K_1\,(n_1,n_2\ge 2)\)

\([\partial _1,\partial _2,(-2)^{n_1+n_2-2}]\)

\([x^2+(4-2(n_1+n_2))x-4(n_1+n_2)+3n_1n_2+4]\)

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Lu, L., Huang, Q. & Huang, X. The graphs with exactly two distance eigenvalues different from \(-1\) and \(-3\) . J Algebr Comb 45, 629–647 (2017). https://doi.org/10.1007/s10801-016-0718-2

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