A Complete Solution to the Cvetkovi\'{c}-Rowlinson Conjecture

In 1990, Cvetkovi\'{c} 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 $n=6$.Comment: Since the conjecture is solved completely now, we change the title into "A Complete Solution to the Cvetkovi\'{c}-Rowlinson Conjecture". 9 page

Spectral extremal results on edge blow-up of graphs

The edge blow-up $F^{p+1}$ of a graph $F$ for an integer $p\geq 2$ is obtained by replacing each edge in $F$ with a $K_{p+1}$ containing the edge, where the new vertices of $K_{p+1}$ are all distinct. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and maximum spectral radius of an $F$-free graph of order $n$, respectively. In this paper, we determine the range of $spex(n,F^{p+1})$ when $F$ is bipartite and the exact value of $spex(n,F^{p+1})$ when $F$ is non-bipartite for sufficiently large $n$, which are the spectral versions of Tur\'{a}n's problems on $ex(n,F^{p+1})$ solved by Yuan [J. Combin. Theory Ser. B 152 (2022) 379--398]. This generalizes several previous results on $F^{p+1}$ for $F$ being a matching, or a star. Additionally, we also give some other interesting results on $F^{p+1}$ for $F$ being a path, a cycle, or a complete graph. To obtain the aforementioned spectral results, we utilize a combination of the spectral version of the Stability Lemma and structural analyses. These approaches and tools give a new exploration of spectral extremal problems on non-bipartite graphs

The largest eigenvalue of C4−\mathcal{C}_4^{-}-free signed graphs

Let $\mathcal{C}_{k}^{-}$ be the set of all negative $C_k$. For odd cycle, Wang, Hou and Li [29] gave a spectral condition for the existence of negative $C_3$ in unbalanced signed graphs. For even cycle, we determine the maximum index among all $\mathcal{C}_4^{-}$-free unbalanced signed graphs and completely characterize the extremal signed graph in this paper. This could be regarded as a signed graph version of the results by Nikiforov [23] and Zhai and Wang [37]

Spectral expansion properties of pseudorandom bipartite graphs

An $(a,b)$-biregular bipartite graph is a bipartite graph with bipartition $(X, Y)$ such that each vertex in $X$ has degree $a$ and each vertex in $Y$ has degree $b$. By the bipartite expander mixing lemma, biregular bipartite graphs have nice pseudorandom and expansion properties when the second largest adjacency eigenvalue is not large. In this paper, we prove several explicit properties of biregular bipartite graphs from spectral perspectives. In particular, we show that for any $(a,b)$-biregular bipartite graph $G$, if the spectral gap is greater than $\frac{2(k-1)}{\sqrt{(a+1)(b+1)}}$, then $G$ is $k$-edge-connected; and if the spectral gap is at least $\frac{2k}{\sqrt{(a+1)(b+1)}}$, then $G$ has at least $k$ edge-disjoint spanning trees. We also prove that if the spectral gap is at least $\frac{(k-1)\max\{a,b\}}{2\sqrt{ab - (k-1)\max\{a,b\}}}$, then $G$ is $k$-connected for $k\ge 2$; and if the spectral gap is at least $\frac{6k+2\max\{a,b\}}{\sqrt{(a-1)(b-1)}}$, then $G$ has at least $k$ edge-disjoint spanning 2-connected subgraphs. We have stronger results in the paper

Sharp upper bounds on the distance spectral radius of a graph

AbstractLet M=(mij) be a nonnegative irreducible n×n matrix with diagonal entries 0. The largest eigenvalue of M is called the spectral radius of the matrix M, denoted by ρ(M). In this paper, we give two sharp upper bounds of the spectral radius of matrix M. As corollaries, we give two sharp upper bounds of the distance matrix of a graph

l-connectivity, l-edge-connectivity and spectral radius of graphs

Let G be a connected graph. The toughness of G is defined as t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of components of G-S. Confirming a conjecture of Brouwer, Gu [SIAM J. Discrete Math. 35 (2021) 948--952] proved a tight lower bound on toughness of regular graphs in terms of the second largest absolute eigenvalue. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] then studied the toughness of simple graphs from the spectral radius perspective. While the toughness is an important concept in graph theory, it is also very interesting to study |S| for which c(G-S)\geq l for a given integer l\geq 2. This leads to the concept of the l-connectivity, which is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Gu [European J. Combin. 92 (2021) 103255] discovered a lower bound on the l-connectivity of regular graphs via the second largest absolute eigenvalue. As a counterpart, we discover the connection between the l-connectivity of simple graphs and the spectral radius. We also study similar problems for digraphs and an edge version

Spectral extremal problem on tt copies of ℓ\ell-cycle

Denote by $tC_\ell$ the disjoint union of $t$ cycles of length $\ell$. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and spectral radius over all $n$-vertex $F$-free graphs, respectively. In this paper, we shall pay attention to the study of both $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any integers $t,\ell$ and $n\ge f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This generalizes the result on $ex(n,tC_3)$ of Erd\H{o}s [Arch. Math. 13 (1962) 222--227] as well as the research on $ex(n,C_{2\ell+1})$ of F\"{u}redi and Gunderson [Combin. Probab. Comput. 24 (2015) 641--645]. On the other hand, we focus on the spectral Tur\'{a}n-type function $spex(n,tC_{\ell})$, and determine the extremal graph for any fixed $t,\ell$ and large enough $n$. Our results not only extend some classic spectral extremal results on triangles, quadrilaterals and general odd cycles due to Nikiforov, but also develop the famous spectral even cycle conjecture proposed by Nikiforov (2010) and confirmed by Cioab\u{a}, Desai and Tait (2022).Comment: 25 pages, one figur