Search

In this paper we consider the eigenvalues and the Seidel eigenvalues of a chain graph. An$\dbar$eli\'{c}, da Fonseca, Simi\'{c}, and Du \cite{andelic2020tridiagonal} conjectured that there do not exist non-isomorphic cospectral chain graphs with respect to the adjacency spectrum. Here we disprove this conjecture. Furthermore, by considering the relation between the Seidel matrix and the adjacency matrix of a graph, we solve two problems on the number of distinct Seidel eigenvalues of a chain graph, which was posed by Mandal, Mehatari, and Das \cite{mandal2022spectrum}., Comment: 13 pages, 2 figures

In this paper, we study the Tur\'{a}n problem for $C_{2k+1}^{-}$. Suppose that $\dot{G}$ is an unbalanced signed graph of order $n$ with $e(\dot{G})$ edges. Let $\lambda_{1} (\dot{G})$ be the largest eigenvalue of $\dot{G}$, and $C_{2k+1}^{-}$ be the set of the negative cycle with length $2k+1$($3 \le k \le \frac{n}{15}$). We prove that if $\dot{G}$ is a $C_{2k+1}^{-}$-free unbalanced signed graph, then $e(\dot{G}) \le e(C_{3}^{-} \cdot K_{n-2})$ and $\lambda_{1}(\dot{G}) \le \lambda_{1}(C_{3}^{-} \cdot K_{n-2})$, with equality holding if and only if $\dot{G}$ is switching equivalent to $C_{3}^{-} \cdot K_{n-2}$.

In this paper, we study the Tur\'{a}n problem of signed graphs version. Suppose that $\dot{G}$ is a connected unbalanced signed graph of order $n$ with $e(\dot{G})$ edges and $e^-(\dot{G})$ negative edges, and let $\rho(\dot{G})$ be the spectral radius of $\dot{G}.$ The signed graph $\dot{G}^{s,t}$ ($s+t=n-2$) is obtained from an all-positive clique $(K_{n-2},+)$ with $V(K_{n-2})=\{u_1,\dots,u_s,v_1,\dots,v_t\}$ ($s,t\ge 1$) and two isolated vertices $u$ and $v$ by adding negative edge $uv$ and positive edges $uu_1,\dots,uu_s,vv_1,\dots,vv_t.$ Firstly, we prove that if $\dot{G}$ is $C_3^-$-free, then $e(\dot{G})\le \frac{n(n-1)}{2}-(n-2),$ with equality holding if and only if $\dot{G}\sim \dot{G}^{s,t}.$ Moreover, $e^-(\dot{G}^{s,t})\le \lfloor\frac{n-2}{2}\rfloor\lceil\frac{n-2}{2}\rceil+n-2,$ with equality holding if and only if $\dot{G}^{s,t}= \dot{G}_U^{\lfloor\frac{n-2}{2}\rfloor,\lceil\frac{n-2}{2}\rceil},$ where $\dot{G}_U^{\lfloor\frac{n-2}{2}\rfloor,\lceil\frac{n-2}{2}\rceil}$ is obtained from $\dot{G}^{\lfloor\frac{n-2}{2}\rfloor,\lceil\frac{n-2}{2}\rceil}$ by switching at vertex set $U=\{v,u_1,\dots,u_{\lfloor\frac{n-2}{2}\rfloor}\}.$ Secondly, we prove that if $\dot{G}$ is $C_3^-$-free, then $\rho(\dot{G})\le \frac{1}{2}( \sqrt{ n^2-8}+n-4),$ with equality holding if and only if $\dot{G}\sim \dot{G}^{1,n-3}.$, Comment: 19 pages, 1 figures

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. The $r$-expansion $F^{r}$ of a graph $F$ is the $r$-uniform hypergraph obtained from $F$ by enlarging each edge of $F$ with a vertex subset of size $r-2$ disjoint from the vertex set of $F$ such that distinct edges are enlarged by disjoint subsets. Let $ex_{r}^{lin}(n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ be the maximum number of edges and the maximum spectral radius of all $F^{r}$-free linear $r$-uniform hypergraphs with $n$ vertices, respectively. In this paper, we present the sharp (or asymptotic) bounds of $ex_{r}^{lin}( n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ by establishing the connection between the spectral radii of linear hypergraphs and those of their shadow graphs, where $F$ is a $(k+1)$-color critical graph or a graph with chromatic number $k$.

In 2001, Davies, Gladwell, Leydold, and Stadler proved discrete nodal domain theorems for eigenfunctions of generalized Laplacians. In 2019, Jost and Mulas generalized the normalized combinatorial Laplace operator of graphs to signed hypergraphs. In this paper, we establish nodal domain theorems for the normalized combinatorial Laplace operator in signed hypergraphs. We also obtain a lower bound estimates for the number of strong nodal domains., Comment: 25 pages, 1 figures. arXiv admin note: substantial text overlap with arXiv:2201.00198 by other authors

The Hoffman program with respect to any real or complex square matrix $M$ associated to a graph $G$ stems from Hoffman's pioneering work on the limit points for the spectral radius of adjacency matrices of graphs does not exceed $\sqrt{2+\sqrt{5}}$. A signed graph $\dot{G}=(G,\sigma)$ is a pair $(G,\sigma),$ where $G=(V,E)$ is a simple graph and $\sigma: E(G)\rightarrow \{+1,-1\}$ is the sign function. In this paper, we study the Hoffman program of signed graphs. Here, all signed graphs whose spectral radius does not exceed $\sqrt{2+\sqrt{5}}$ will be identified., Comment: 33 pages, 20 figures

Let $x_1, x_2, \dots, x_n$ be the eigenvalues of a signed graph $\Gamma$ of order $n$. The energy of $\Gamma$ is defined as $E(\Gamma)=\sum^{n}_{j=1}|x_j|.$ Let $\mathcal{P}_n^4$ be obtained by connecting a vertex of the negative circle $(C_4,{\overline{\sigma}})$ with a terminal vertex of the path $P_{n-4}$. In this paper, we show that for $n=4,6$ and $n \geq 8,$ $\mathcal{P}_n^4$ has the maximal energy among all connected unicyclic $n$-vertex signed graphs, except the cycles $C_5^+, C_7^+.$

In this paper, all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $2$ and $-1$ are determined. These graphs conclude a class of generalized friendship graphs $F_{t,r,k}, $ which is the graph of $k$ copies of the complete graph $K_t$ meeting in common $r$ vertices such that $t-r=3.$ Which of these graphs are determined by its spectrum is are also obtained., Comment: 18 pages, 2 figures. arXiv admin note: text overlap with arXiv:1310.6529 by other authors

Tittmann, Averbouch and Makowsky [P. Tittmann, I. Averbouch, J.A. Makowsky, The enumeration of vertex induced subgraphs with respect to the number of components, European Journal of Combinatorics, 32 (2011) 954-974], introduced the subgraph component polynomial $Q(G;x,y)$ which counts the number of connected components in vertex induced subgraphs. It contains much of the underlying graph's structural information, e.g. the order, the size, the independence number. We show that there are several other graph invariants, like the connectivity, the number of cycles of length four in a regular bipartite graph, are determined by the subgraph component polynomial. Using the obtained results, we find several well-known families of graphs are determined by $Q(G;x,y)$: paths, cycles, tadpole graphs, complete bipartite graphs, friendship graphs, book graphs and hypercubes. Moreover, we study the distinguish power and find some simple graphs which are not distinguished by the subgraph component polynomial but distinguished by the Tutte polynomial and the character polynomial. These are answers to three questions concerning the subgraph component polynomial proposed by Tittmann et al., Comment: 13 pages, 5 figures

In this paper, we find recursive formulas for the Tutte polynomial of a family of small-world networks: Farey graphs, which are modular and have an exponential degree hierarchy. Then, making use of these formulas, we determine the number of spanning trees, as well as the number of connected spanning subgraphs. Furthermore, we also derive exact expressions for the chromatic polynomial and the reliability polynomial of these graphs., Comment: 6 pages