arXiv subjects
Publications and source records attributed to Huiqiu Lin.
In this paper we prove that balanced characteristic functions of canonical cliques in a Paley graph of square order $P(q^2)$ span the $\frac{-1+q}{2}$-eigenspace of the graph. This is the first of two steps to a second proof of the analogue of Erdös-Ko-Rado theorem for Paley graphs of square order (the first proof was given by A. Blokhuis in 1984).
Nikiforov [Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 179--189] showed that if $G$ is $K_{r+1}$-free then the spectral radius $ρ(G)\leq\sqrt{2m(1-1/r)}$, which implies that $G$ contains $C_3$ if $ρ(G)>\sqrt{m}$. In this paper, we follow this direction on determining which subgraphs will be contained in $G$ if $ρ(G)> f(m)$, where $f(m)\sim\sqrt{m}$ as $m\rightarrow \infty$. We first show that if $ρ(G)\geq \sqrt{m}$, 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 $ρ(G)\geq{\frac12+\sqrt{m-\frac34}}$, then $G$ contains pentagon and hexagon unless $G$ is a book; and if $ρ(G)>{\frac12(k-\frac12)+\sqrt{m+\frac14(k-\frac12)^2}}$, then $G$ contains $C_t$ for every $t\leq 2k+2$. In the end, some related conjectures are provided for further research.
A perfect matching in a graph $G$ is a set of nonadjacent edges covering every vertex of $G$. Motivated by recent progress on the relations between the eigenvalues and the matching number of a graph, in this paper, we aim to present a distance spectral radius condition to guarantee the existence of a perfect matching. Let $G$ be an $n$-vertex connected graph where $n$ is even and $λ_{1}(D(G))$ be the distance spectral radius of $G$. Then the following statements are true. \noindent$\rm{I)}$ If $4\le n\le10$ and ${λ}_{1} (D\left(G\right))\le {λ}_{1} (D(S_{n,{\frac{n}{2}}-1}))$, then $G$ contains a perfect matching unless $G\cong S_{n,{\frac{n}{2}-1}}$ where $S_{n,{\frac{n}{2}-1}}\cong K_{{\frac{n}{2}-1}}\vee ({\frac{n}{2}+1})K_1$. \noindent$\rm{II)}$ If $n\ge 12$ and ${λ}_{1} (D\left(G\right))\le {λ}_{1} (D(G^*))$, then $G$ contains a perfect matching unless $G\cong G^*$ where $G^*\cong K_1\vee (K_{n-3}\cup2K_1)$. Moreover, if $G$ is a connected $2n$-vertex balanced bipartite graph with $λ_{1}(D(G))\le λ_{1}(D(B_{n-1,n-2})) $, then $G$ contains a perfect matching, unless $G\cong B_{n-1,n-2}$ where $B_{n-1,n-2}$ is obtained from $K_{n,n-2}$ by attaching two pendent vertices to a vertex in the $n$-vertex part.
In 1990, Cvetković and Rowlinson [The largest eigenvalue of a graph: a survey, Linear Multilinear Algebra 28(1-2) (1990), 3--33] conjectured that among all outerplanar graphs on $n$ vertices, $K_1\vee P_{n-1}$ attains the maximum spectral radius. In 2017, Tait and Tobin [Three conjectures in extremal spectral graph theory, J. Combin. Theory, Ser. B 126 (2017) 137-161] confirmed the conjecture for sufficiently large values of $n$. In this article, we show the conjecture is true for all $n\geq2$ except for $n=6$.
A wheel graph is a graph formed by connecting a single vertex to all vertices of a cycle. A graph is called wheel-free if it does not contain any wheel graph as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Turán type problem: what is the maximum spectral radius of a graph of order $n$ that does not contain subgraphs of particular kind. In this paper, we study the Brualdi-Solheid-Turán type problem for wheel-free graphs, and we determine the maximum (signless Laplacian) spectral radius of a wheel-free graph of order $n$. Furthermore, we characterize the extremal graphs.
Bollobás and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $λ^2_1(G)+λ^2_2(G)\leq \frac{r-1}{r}\cdot2m$, where $λ_1(G)$ and $λ_2(G)$ are the largest and the second largest eigenvalues of the adjacency matrix $A(G)$, respectively. In this paper, we confirm the conjecture in the case $r=2$, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdős and Nosal respectively, we prove that every non-bipartite graph $G$ of order $n$ and size $m$ contains a triangle, if one of the following is true: (1) $λ_1(G)\geq\sqrt{m-1}$ and $G\neq C_5\cup (n-5)K_1$; and (2) $λ_1(G)\geq λ_1(S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}))$ and $G\neq S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil})$, where $S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil})$ is obtained from $K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.
Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $α\in [0,1]$, Nikiforov \cite{VN1} defined the matrix $A_α(G)$ as $$A_α(G)=αD(G)+(1-α)A(G).$$ In this paper, we give some results on the eigenvalues of $A_α(G)$ with $α>1/2$. In particular, we show that for each $e\notin E(G)$, $λ_i(A_α(G+e))\geqλ_i(A_α(G))$. By utilizing the result, we prove have $λ_k(A_α(G))\leqαn-1$ for $2\leq k\leq n$. Moreover, we characterize the extremal graphs with equality holding. Finally, we show that $λ_n(A_α({G}))\geq 2α-1$ if $G$ contains no isolated vertices.
Let $k\ge 2$ be an integer. If a square 0-1 matrix $A$ satisfies $A^k=A$, then $A$ is said to be $k$-idempotent. In this paper, we give a characterization of $k$-idempotent 0-1 matrices. We also determine the maximum number of nonzero entries in $k$-idempotent 0-1 matrices of a given order as well as the $k$-idempotent 0-1 matrices attaining this maximum number.
Let $G$ be a graph of order $n$, and let $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)$ denote the signless Laplacian eigenvalues of $G$. Ashraf and Tayfeh-Rezaie [Electron. J. Combin. 21 (3) (2014) \#P3.6] showed that $q_1(G)+q_1(\overline{G})\leq 3n-4$, with equality holding if and only if $G$ or $\overline{G}$ is the star $K_{1,n-1}$. In this paper, we discuss the following problem: for $n\geq6$, does $q_2(G)+q_2(\overline{G})\leq 2n-5$ always hold? We provide positive answers to this problem for the graphs with disconnected complements and the bipartite graphs, and determine the graphs attaining the bound. Moreover, we show that $q_2(G)+q_2(\overline{G})\geq n-2$, and the extremal graphs are also characterized.
For a connected graph $G$ with order $n$ and an integer $k\geq 1$, we denote by $$S_k(D(G))=λ_1(D(G))+\cdots+λ_k(D(G))$$ the sum of $k$ largest distance eigenvalues of $G$. In this paper, we consider the sharp upper bound and lower bound of $S_k(D(G))$. We determine the sharp lower bounds of $S_k(D(G))$ when $G$ is connected graph and is a tree, respectively, and characterize both the extremal graphs. Moreover, we conjecture that the upper bound is attained when $G$ is a path of order $n$ and prove some partial result supporting the conjecture. To prove our result, we obtain a sharp upper bound of $λ_2(D(G))$ in terms of the order and the diameter of $G$, where $λ_2(D(G))$ is the second largest distance eigenvalue of $G$. As applications, we prove a general inequality involving $λ_2(D(G))$, the independence number of $G$, and the number of triangles in $G$. An immediate corollary is a conjecture of Fajtlowicz, which was confirmed in \cite{L15-L} by a different argument. We conclude this paper with some open problems for further study.
Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $α\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107] defined the matrix $A_α(G)$ as $A_α(G)=αD(G)+(1-α)A(G).$ Let $u$ and $v$ be two vertices of a connected graph $G$. Suppose that $u$ and $v$ are connected by a path $w_0(=v)w_1\cdots w_{s-1}w_s(=u)$ where $d(w_i)=2$ for $1\leq i\leq s-1$. Let $G_{p,s,q}(u,v)$ be the graph obtained by attaching the paths $P_p$ to $u$ and $P_q$ to $v$. Let $s=0,1$. Nikiforov and Rojo [On the $α$-index of graphs with pendent paths, Linear Algebra Appl. 550 (2018) 87--104] conjectured that $ρ_α(G_{p,s,q}(u,v))<ρ_α(G_{p-1,s,q+1}(u,v))$ if $p\geq q+2.$ In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal $A_α$-spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal $A_α$-spectral radius with fixed order and matching number. These results generalize some known results.
Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_α(G)=αD(G)+(1-α)A(G) $$ for any real $α\in [0,1]$. The collection of eigenvalues of $A_α(G)$ together with multiplicities are called the \emph{$A_α$-spectrum} of $G$. A graph $G$ is said to be \emph{determined by its $A_α$-spectrum} if all graphs having the same $A_α$-spectrum as $G$ are isomorphic to $G$. We first prove that some graphs are determined by its $A_α$-spectrum for $0\leqα<1$, including the complete graph $K_m$, the star $K_{1,n-1}$, the path $P_n$, the union of cycles and the complement of the union of cycles, the union of $K_2$ and $K_1$ and the complement of the union of $K_2$ and $K_1$, and the complement of $P_n$. Setting $α=0$ or $\frac{1}{2}$, those graphs are determined by $A$- or $Q$-spectra. Secondly, when $G$ is regular, we show that $G$ is determined by its $A_α$-spectrum if and only if the join $G\vee K_m$ is determined by its $A_α$-spectrum for $\frac{1}{2}<α<1$. Furthermore, we also show that the join $K_m\vee P_n$ is determined by its $A_α$-spectrum for $\frac{1}{2}<α<1$. In the end, we pose some related open problems for future study.
Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$ and $D(G)$ be the distance matrix of $G$. The distance Laplacian matrix of $G$ is defined as $\mathcal{L}(G)=Tr(G)-D(G)$. The distance signless Laplacian matrix of $G$ is defined as $\mathcal{Q}(G)=Tr(G)+D(G)$. In this paper, we give a lower bound on the distance Laplacian spectral radius in terms of $D_1$, as a consequence, we show that $\partial_1^L(G)\geq n+\lceil\frac{n}ω\rceil$ where $ω$ is the clique number of $G$. Furthermore, we give some graft transformations, by using them, we characterize the extremal graph attains the maximum distance spectral radius in terms of $n$ and $ω$. Moreover, we also give bounds on the distance signless Laplacian eigenvalues of $G$, and give a confirmation on a conjecture due to Aouchiche and Hansen.
In this paper, we consider the distance spectra of the derangement graphs. First we give a constructive proof that the connected derangement graphs are of diameter 2. Then we obtain their distance spectra. In particular, we determine all their extremal distance eigenvalues.
Let $G$ be a connected graph of order $n$ with diameter $d$. Remoteness $ρ$ of $G$ is the maximum average distance from a vertex to all others and $\partial_1\geq\cdots\geq \partial_n$ are the distance eigenvalues of $G$. In \cite{AH}, Aouchiche and Hansen conjectured that $ρ+\partial_3>0$ when $d\geq 3$ and $ρ+\partial_{\lfloor\frac{7d}{8}\rfloor}>0.$ In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on $\partial_n+ρ$ and $\partial_1-ρ$ when $G\ncong K_n$ and the extremal graphs are characterized.
Let $G$ be a graph with $n$ vertices and $λ_n(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal graphs. This result gives spectral conditions for the existence of specified paths and cycles in graphs.