arXiv subjects

Yongtang Shi

Publications and source records attributed to Yongtang Shi.

83 records · Page 5Linked to original sources

arXiv2011

On the maximal energy tree with two maximum degree vertices

For a simple graph $G$, the energy $E(G)$ is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix. For $Δ\geq 3$ and $t\geq 3$, denote by $T_a(Δ,t)$ (or simply $T_a$) the tree formed from a path $P_t$ on $t$ vertices by attaching $Δ-1$ $P_2$'s on each end of the path $P_t$, and $T_b(Δ, t)$ (or simply $T_b$) the tree formed from $P_{t+2}$ by attaching $Δ-1$ $P_2$'s on an end of the $P_{t+2}$ and $Δ-2$ $P_2$'s on the vertex next to the end. In [X. Li, X. Yao, J. Zhang and I. Gutman, Maximum energy trees with two maximum degree vertices, J. Math. Chem. 45(2009), 962--973], Li et al. proved that among trees of order $n$ with two vertices of maximum degree $Δ$, the maximal energy tree is either the graph $T_a$ or the graph $T_b$, where $t=n+4-4Δ\geq 3$. However, they could not determine which one of $T_a$ and $T_b$ is the maximal energy tree. This is because the quasi-order method is invalid for comparing their energies. In this paper, we use a new method to determine the maximal energy tree. It turns out that things are more complicated. We prove that the maximal energy tree is $T_b$ for $Δ\geq 7$ and any $t\geq 3$, while the maximal energy tree is $T_a$ for $Δ=3$ and any $t\geq 3$. Moreover, for $Δ=4$, the maximal energy tree is $T_a$ for all $t\geq 3$ but $t=4$, for which $T_b$ is the maximal energy tree. For $Δ=5$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t$ is odd and $3\leq t\leq 89$, for which $T_a$ is the maximal energy tree. For $Δ=6$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t=3,5,7$, for which $T_a$ is the maximal energy tree. One can see that for most $Δ$, $T_b$ is the maximal energy tree, $Δ=5$ is a turning point, and $Δ=3$ and 4 are exceptional cases.

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arXiv2011

Complete solution to a conjecture on the maximal energy of unicyclic graphs

For a given simple graph $G$, the energy of $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $P_n^{\ell}$ be the unicyclic graph obtained by connecting a vertex of $C_\ell$ with a leaf of $P_{n-\ell}$\,. In [G. Caporossi, D. Cvetković, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, {\it J. Chem. Inf. Comput. Sci.} {\bf 39}(1999) 984--996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is $C_n$ if $n\leq 7$ and $n=9,10,11,13,15$\,, and $P_n^6$ for all other values of $n$. In this paper, by employing the Coulson integral formula and some knowledge of real analysis, especially by using certain combinatorial technique, we completely solve this conjecture. However, it turns out that for $n=4$ the conjecture is not true, and $P_4^3$ should be the unicyclic graph with maximal energy.

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arXiv2011

Solution to a conjecture on the maximal energy of bipartite bicyclic graphs

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $C_n$ denote the cycle of order $n$ and $P^{6,6}_n$ the graph obtained from joining two cycles $C_6$ by a path $P_{n-12}$ with its two leaves. Let $\mathscr{B}_n$ denote the class of all bipartite bicyclic graphs but not the graph $R_{a,b}$, which is obtained from joining two cycles $C_a$ and $C_b$ ($a, b\geq 10$ and $a \equiv b\equiv 2\, (\,\textmd{mod}\, 4)$) by an edge. In [I. Gutman, D. Vidović, Quest for molecular graphs with maximal energy: a computer experiment, {\it J. Chem. Inf. Sci.} {\bf41}(2001), 1002--1005], Gutman and Vidović conjectured that the bicyclic graph with maximal energy is $P^{6,6}_n$, for $n=14$ and $n\geq 16$. In [X. Li, J. Zhang, On bicyclic graphs with maximal energy, {\it Linear Algebra Appl.} {\bf427}(2007), 87--98], Li and Zhang showed that the conjecture is true for graphs in the class $\mathscr{B}_n$. However, they could not determine which of the two graphs $R_{a,b}$ and $P^{6,6}_n$ has the maximal value of energy. In [B. Furtula, S. Radenković, I. Gutman, Bicyclic molecular graphs with the greatest energy, {\it J. Serb. Chem. Soc.} {\bf73(4)}(2008), 431--433], numerical computations up to $a+b=50$ were reported, supporting the conjecture. So, it is still necessary to have a mathematical proof to this conjecture. This paper is to show that the energy of $P^{6,6}_n$ is larger than that of $R_{a,b}$, which proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, the conjecture is still open.

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arXiv2011

The complexity of determining the rainbow vertex-connection of graphs

A vertex-colored graph is {\it rainbow vertex-connected} if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In this paper, we study the computational complexity of vertex-rainbow connection of graphs and prove that computing $rvc(G)$ is NP-Hard. Moreover, we show that it is already NP-Complete to decide whether $rvc(G)=2$. We also prove that the following problem is NP-Complete: given a vertex-colored graph $G$, check whether the given coloring makes $G$ rainbow vertex-connected.

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arXiv2010

Note on a relation between Randic index and algebraic connectivity

A conjecture of AutoGraphiX on the relation between the Randić index $R$ and the algebraic connectivity $a$ of a connected graph $G$ is: $$\frac R a\leq (\frac{n-3+2\sqrt{2}}{2})/(2(1- \cos {\fracπ{n}})) $$ with equality if and only if $G$ is $P_n$, which was proposed by Aouchiche and Hansen [M. Aouchiche and P. Hansen, A survey of automated conjectures in spectral graph theory, {\it Linear Algebra Appl.} {\bf 432}(2010), 2293--2322]. We prove that the conjecture holds for all trees and all connected graphs with edge connectivity $κ'(G)\geq 2$, and if $κ'(G)=1$, the conjecture holds for sufficiently large $n$. The conjecture also holds for all connected graphs with diameter $D\leq \frac {2(n-3+2\sqrt{2})}{π^2}$ or minimum degree $δ\geq \frac n 2$. We also prove $R\cdot a\geq \frac {8\sqrt{n-1}}{nD^2}$ and $R\cdot a\geq \frac {nδ(2δ-n+2)} {2(n-1)}$, and then $R\cdot a$ is minimum for the path if $D\leq (n-1)^{1/4}$ or $δ\geq \frac n 2-1$.

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arXiv2010

On the rainbow vertex-connection

A vertex-colored graph is {\it rainbow vertex-connected} if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. Krivelevich and Yuster proved that if $G$ is a graph of order $n$ with minimum degree $δ$, then $rvc(G)<11n/δ$. In this paper, we show that $rvc(G)\leq 3n/(δ+1)+5$ for $δ\geq \sqrt{n-1}-1$ and $n\geq 290$, while $rvc(G)\leq 4n/(δ+1)+5$ for $16\leq δ\leq \sqrt{n-1}-2$ and $rvc(G)\leq 4n/(δ+1)+C(δ)$ for $6\leqδ\leq 15$, where $C(δ)=e^{\frac{3\log(δ^3+2δ^2+3)-3(\log 3-1)} {δ-3}}-2$. We also prove that $rvc(G)\leq 3n/4-2$ for $δ=3$, $rvc(G)\leq 3n/5-8/5$ for $δ=4$ and $rvc(G)\leq n/2-2$ for $δ=5$. Moreover, an example shows that when $δ\geq \sqrt{n-1}-1$ and $δ=3,4,5$, our bounds are seen to be tight up to additive factors.

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arXiv2010

Complete solution to a problem on the maximal energy of unicyclic bipartite graphs

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of $C_6$ with a leaf of $P_{n-6}$\,. Caporossi et al. conjecture that the unicyclic graph with maximal energy is $P_n^6$ for $n=8,12,14$ and $n\geq 16$. In``Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, {\it Linear Algebra Appl.} {\bf 356}(2002), 27--36", the authors proved that $E(P_n^6)$ is maximal within the class of the unicyclic bipartite $n$-vertex graphs differing from $C_n$\,. And they also claimed that the energy of $C_n$ and $P_n^6$ is quasi-order incomparable and left this as an open problem. In this paper, by utilizing the Coulson integral formula and some knowledge of real analysis, especially by employing certain combinatorial techniques, we show that the energy of $P_n^6$ is greater than that of $C_n$ for $n=8,12,14$ and $n\geq 16$, which completely solves this open problem and partially solves the above conjecture.

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arXiv2010

Rainbow connection in $3$-connected graphs

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper, we proved that $rc(G)\leq 3(n+1)/5$ for all $3$-connected graphs.

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arXiv2009

Note on the 4- and 5-leaf powers

Motivated by the problem of reconstructing evolutionary history, Nishimura et al. defined $k$-leaf powers as the class of graphs $G=(V,E)$ which has a $k$-leaf root $T$, i.e., $T$ is a tree such that the vertices of $G$ are exactly the leaves of $T$ and two vertices in $V$ are adjacent in $G$ if and only if their distance in $T$ is at most $k$. It is known that leaf powers are chordal graphs. Brandstädt and Le proved that every $k$-leaf power is a $(k+2)$-leaf power and every 3-leaf power is a $k$-leaf power for $k\geq 3$. They asked whether a $k$-leaf power is also a $(k+1)$-leaf power for any $k\geq 4$. Fellows et al. gave an example of a 4-leaf power which is not a 5-leaf power. It is interesting to find all the graphs which have both 4-leaf roots and 5-leaf roots. In this paper, we prove that, if $G$ is a 4-leaf power with $L(G)\neq \emptyset$, then $G$ is also a 5-leaf power, where $L(G)$ denotes the set of leaves of $G$.

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arXiv2009

Note on the energy of regular graphs

For a simple graph $G$, the energy $\mathcal{E}(G)$ is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix $A(G)$. Let $n, m$, respectively, be the number of vertices and edges of $G$. One well-known inequality is that $\mathcal{E}(G)\leq λ_1+\sqrt{(n-1)(2m-λ_1)}$, where $λ_1$ is the spectral radius. If $G$ is $k$-regular, we have $\mathcal{E}(G)\leq k+\sqrt{k(n-1)(n-k)}$. Denote $\mathcal{E}_0=k+\sqrt{k(n-1)(n-k)}$. Balakrishnan [{\it Linear Algebra Appl.} {\bf 387} (2004) 287--295] proved that for each $ε>0$, there exist infinitely many $n$ for each of which there exists a $k$-regular graph $G$ of order $n$ with $k< n-1$ and $\frac{\mathcal{E}(G)}{\mathcal{E}_0}<ε$, and proposed an open problem that, given a positive integer $n\geq 3$, and $ε>0$, does there exist a $k$-regular graph $G$ of order $n$ such that $\frac{\mathcal{E}(G)}{\mathcal{E}_0}>1-ε$. In this paper, we show that for each $ε>0$, there exist infinitely many such $n$ that $\frac{\mathcal{E}(G)}{\mathcal{E}_0}>1-ε$. Moreover, we construct another class of simpler graphs which also supports the first assertion that $\frac{\mathcal{E}(G)}{\mathcal{E}_0}<ε$.

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arXiv2009

Randić index, diameter and the average distance

The Randić index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1/\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randić index $R(G)$ with relations to the diameter $D(G)$ and the average distance $μ(G)$ of a graph $G$. We prove that for any connected graph $G$ of order $n$ with minimum degree $δ(G)$, if $δ(G)\geq 5$, then $R(G)-D(G)\geq \sqrt 2-\frac{n+1} 2$; if $δ(G)\geq n/5$ and $n\geq 15$, $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq μ(G)$. Furthermore, for any arbitrary real number $\varepsilon \ (0<\varepsilon<1)$, if $δ(G)\geq \varepsilon n$, then $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq μ(G)$ hold for sufficiently large $n$.

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