Spectral extrema of graphs with fixed size: Cycles and complete bipartite graphs

Adjacency eigenvalues of graphs without short odd cycles

A spectral version of Mantel's theorem

A spectral condition for odd cycles in non-bipartite graphs

This survey is two-fold. We first report new progress on the spectral extremal results on the Tur\'{a}n type problems in graph theory. More precisely, we shall summarize the spectral Tur\'{a}n function in terms of the adjacency spectral radius and the signless Laplacian spectral radius for various graphs. For instance, the complete graphs, general graphs with chromatic number at least three, complete bipartite graphs, odd cycles, even cycles, color-critical graphs and intersecting triangles. The second goal is to conclude some recent results of the spectral conditions on some graphical properties. By a unified method, we present some sufficient conditions based on the adjacency spectral radius and the signless Laplacian spectral radius for a graph to be Hamiltonian, $k$-Hamiltonian, $k$-edge-Hamiltonian, traceable, $k$-path-coverable, $k$-connected, $k$-edge-connected, Hamilton-connected, perfect matching and $\beta$-deficient.

A survey on spectral conditions for some extremal graph problems

Bollobas and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.

pdf/eigenvalues-and-triangles-in-graphs-4tx749ht40.pdf

Eigenvalues and triangles in graphs

A complete solution to the Cvetković–Rowlinson conjecture

Bollob\'as 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 $\lambda^2_1(G)+\lambda^2_2(G)\leq \frac{r-1}{r}\cdot2m$, where $\lambda_1(G)$ and $\lambda_2(G)$ 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 Erd\H{o}s 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) $\lambda_1(G)\geq\sqrt{m-1}$ and $G\neq C_5\cup (n-5)K_1$; and (2) $\lambda_1(G)\geq \lambda_1(S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}))$ 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.

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 $n=6$.

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

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Eigenvalues and triangles in graphs

A complete solution to the Cvetković–Rowlinson conjecture

Eigenvalues and triangles in graphs

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