For a set of graphs F, a graph G is called F-free if it does not contain any graph in F as a subgraph. Let SPEX(n, F) denote the graphs with the maximum spectral radius among all F-free graphs of order n. In this paper, for any non-bipartite graph H, we give some characterizations for the graphs in SPEX(n, {& xcup;(l)(i=1) P-ki, H}) for sufficiently large n, where l >= 1, k(1) >= >= k(l) >= 2, and there exists at least one ki not equal to 3. As an application, we completely characterize the graphs in SPEX(n, {& xcup;(l)(i=1) P-ki, Kk+1}) and SPEX(n, {& xcup;(l)(i=1) P-ki, F-k}) for sufficiently large n, where F-k is a friendship graph on 2k+ 1 vertices consisting of k triangles which intersect in exactly one common vertex.

Let F be a family of graphs. A graph is called F-free if it does not contain any member of F as a subgraph. A star forest Us+1 (i=1) S-di is a forest whose all components are stars, where S-di is a star of order d(i) + 1. In this paper, for k >= 2, s >= 1 and n >= 100 (Sigma(s+1)(i=1) d(i) + 3s + 1 )(6) , we obtain the maximum spectral radius of {Kk+1, Us+1 (i=1) S-di }-free graphs of order n, where d(1) >= ... >= d(s+1) >= 1. Moreover, we also characterize the extremal graphs. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let F be a graph with chromatic number χ(F) = r+1. Denote by ex(n, F) and Ex(n, F) the Turán number and the set of all extremal graphs for F, respectively. In addition, ex_ssp(n, F) and Ex_ssp(n, F) are the maximum signless Laplacian spectral radius of all n-vertex F-free graphs and the set of all n-vertex F-free graphs with signless Laplacian spectral radius ex_ssp(n, F), respectively. It is known that Ex_ssp(n, F)⊃ Ex(n, F) if F is a triangle. In this paper, employing the regularity method and Füredi's stability theorem, we prove that for a given graph F and r⩾ 3, if ex(n, F) = t_r(n)+O(1), then Ex_ssp(n, F) ⊆ Ex(n, F) for sufficiently large n, where t_r(n) is the number of edges in the Turán graph T_r(n).

Given a graph H, a graph is called H-free if it does not contain Has a subgraph. For a positive integer k, a book Bk+1 is a graph consisting of k + 1 triangles sharing a common edge. In this paper, let G be a Bk+1-free graph of order n >= 49k2 -22k+4. Then the signless ,/ Laplacian spectral radius q(G) <= n+2k+(n-2k)2+8k2 2 with equality if and only if G = Kk boolean OR H, where k or n - k is even, and His a K3-free k-regular graph of order n - k. Furthermore, if k and n - k are both odd, the extremal graphs with the maximum signless Laplacian spectral radius are also characterized. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The Gruenberg-Kegel graph (or the prime graph) Γ(G) of a finite group G is the graph whose vertex set is the set of prime divisors of |G| and in which two distinct vertices r and s are adjacent if and only if there exists an element of order rs in G. A group G is recognizable by isomorphism type of Gruenberg–Kegel graph if for every group H the isomorphism between Γ(H) and Γ(G) as abstract graphs (i. e. unlabeled graphs) implies that G≅ H. In this paper, we prove that finite simple exceptional groups of Lie type ^2E_6(2) and E_8(q) for q ∈{3, 4, 5, 7, 8, 9, 17} are recognizable by isomorphism type of Gruenberg-Kegel graph.

Let $F_{a_1,\dots,a_k}$ be a graph consisting of $k$ cycles of odd length $2a_1+1,\dots, 2a_k+1,$ respectively, which intersect in exactly one common vertex, where $k\geq1$ and $a_1\ge a_2\ge \cdots\ge a_k\ge 1$. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all $F_{a_1,\dots,a_k}$-free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.

If $G$ is a finite group, then the spectrum $\omega(G)$ is the set of all element orders of $G$. The prime spectrum $\pi(G)$ is the set of all primes belonging to $\omega(G)$. A simple graph $\Gamma(G)$ whose vertex set is $\pi(G)$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if $rs \in \omega(G)$ is called the Gruenberg-Kegel graph or the prime graph of $G$. In this paper, we prove that if $G$ is a group of even order, then the set of vertices which are non-adjacent to $2$ in $\Gamma(G)$ form a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg-Kegel graph of a finite group. Besides this, we prove that a complete bipartite graph with each part of size at least $3$ can not be isomorphic to the Gruenberg-Kegel graph of a finite group.

In this paper, we present a sharp upper bound for the spectral radius of an n-vertex graph without F-minor for sufficient large n, where F is obtained from the complete graph K_r by deleting disjointed paths. Furthermore, the graphs which achieved the sharp bound are characterized. This result may be regarded to be an extended revision of the number of edges in an n-vertex graph without F-minor.

In this paper, we present two sharp upper bounds for the spectral radius of (bipartite) graphs with forbidden a star forest and characterize all extremal graphs. Moreover, the minimum least eigenvalue of the adjacency matrix of graph with forbidden a star forest and all extremal graphs for graphs are obtained.

The bipartite Turán number of a graph $H$, denoted by $ex(m,n; H)$, is the maximum number of edges in any bipartite graph $G=(X,Y; E)$ with $|X|=m$ and $|Y|=n$ which does not contain $H$ as a subgraph. In this paper, we determined $ex(m,n; F_{\ell})$ for arbitrary $\ell$ and appropriately large $n$ with comparing to $m$ and $\ell$, where $F_\ell$ is a linear forest which consists of $\ell$ vertex disjoint paths. Moreover, the extremal graphs have been characterized. Furthermore, these results are used to obtain the maximum spectral radius of bipartite graphs which does not contain $F_{\ell}$ as a subgraph and characterize all extremal graphs which attain the maximum spectral radius.

Let sigma be a partition of the set of prime numbers. In this paper, we describe the finite groups for which every sigma -subnormal subgroup is modular.

In this paper, we determine the maximum signless Laplacian spectral radius of all graphs which do not contain small books as a subgraph and characterize all extremal graphs. In addition, we give an upper bound of the signless Laplacian spectral radius of all graphs which do not contain intersecting quadrangles as a subgraph.

Let A be a subgroup of a finite group G and 𝒳 = {X_ 1, … , X_ t} a set of subgroups of G. Then we say that 𝒳 is a covering subgroup system for A in G if A=⟨ X _1∩ A, … , X _t∩ A ⟩ and (|X _i|, |X _j|)=1 for all i j . In this paper, we generalize some results of the theory of permutable subgroups in Skiba (Algebra 436:1–16, 2015), Ore (Duke Math J 5:431–460, 1939), Ito and Szep (Act Sci Math 23:168–170, 1962), Deskins (Math Z 82:125–132, 1963), Kegel (Math Z 78:20–221, 1962), Isaacs (Arch Math 102:1–6, 20140, Guo and Skiba (Monatsh Math 185:443–453, 2018). In particular, we prove the following result. Theorem. Let A be a subgroup of a finite group G and E=(X_ 1⋯ X_ t)^G , where 𝒳 = {X_ 1, … , X_ t} is a covering subgroup system for A in G. Suppose that AX_i^x=X_i^xA for all i and all x∈ G . If every member of 𝒳 is a soluble (respectively primary) group, then the section A^E /A_ E is soluble (respectively nilpotent).

Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph of order $n$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. In this paper, we present a sharp upper bound for the signless spectral radius of $G$ without any tree and characterize all extremal graphs which attain the upper bound, which may be regarded as a spectral extremal version for the famous Erd\H{o}s-Sós conjecture.

Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_\alpha (G)$ of $A (G)$ and $D (G) $ by $$A_\alpha (G)=\alpha D(G)+(1-\alpha)A(G)$$ for any real number $0\leq\alpha\leq1$. The \emph{$\alpha$-index} of $G$ is the largest eigenvalue of $A_\alpha(G)$. In this paper, we determine the maximum $\alpha$-index and characterize all extremal graphs for $K_r$ minor-free graphs, $K_{s,t}$ minor-free graphs, and star-forest-free graphs for any $0<\alpha<1$ by unified eigenvector approach, respectively.

Turán type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Turán type extremal problem is how to maximize (signless Laplacian) spectral radius over all graphs which do not contain fixed subgraphs. In this paper, we first present a stability result for k⋅P3 in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or k⋅P3 as a subgraph, respectively.

The Turán type extremal problems ask to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, their spectral counterparts ask to maximize spectral radius of all graphs which do not contain fixed subgraphs. In this paper, we determine the maximum spectral radius of all graphs without a linear forest as a subgraph and all the extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without \(k\cdot P_3\) as a subgraph are obtained and all the extremal graphs are also determined. Moreover, some relations between Turán type extremal problems and their spectral counterparts are discussed.

A balanced bipartite graph G is said to be 2p-Hamilton-biconnected if for any balanced subset W of size 2p of V(G), the subgraph induced by V(G)nW is Hamilton-biconnected. In this paper, we prove that ?Let G be a balanced bipartite graph of order 2n with minimum degree ?(G) ? k, where n ? 2k-p+2 for two integers k ? p ? 0. If the number of edges e(G) > n(n-k + p-1) + (k + 2)(k-p+1), then G is 2p-Hamilton-biconnected except some exceptions.? Furthermore, this result is used to present two new spectral conditions for a graph to be 2p-Hamilton-biconnected. Moreover, the similar results are also presented for nearly balanced bipartite graphs.

The Erdős–Gallai Theorem states that every graph of average degree more than l−2 contains a path of order l for l≥2. In this paper, we obtain a stability version of the Erdős–Gallai Theorem in terms of minimum degree. Let G be a connected graph of order n and F=(⋃i=1kP2ai)⋃(⋃i=1lP2bi+1) be k+l disjoint paths of order 2a1,…,2ak,2b1+1,…,2bl+1, respectively, where k≥0, 0≤l≤2, and k+l≥2. If the minimum degree δ(G)≥∑i=1kai+∑i=1lbi−1, then F⊆G except several classes of graphs for sufficiently large n, which extends and strengths the results of Ali and Staton for an even path and Yuan and Nikiforov for an odd path.

This paper systematically introduced some new results and problems in spectral extremal graph theory.It contained many results of a variety of Turán types,including complete subgraphs,linear forests,circles,bipartite graphs,and minors with respect to adjacency spectrum and signless Laplacian spectrum.Some conjectures and problems were also included.