We study the random variable that counts the number of specific arrangements of clustered consecutive numbers in permutations under the Mallows distribution. We provide an asymptotic expression for the expected value of this random variable. This result extends and tightens the previously known result by Pinsky (2022) concerning clustered consecutive numbers in Mallows permutations. Moreover, we identify a range of parameters for which the distribution of the number of arrangements of clustered consecutive numbers in Mallows permutations is close to a Poisson distribution.

A family of $m \times n$ matrices $\mathcal{F} \subseteq \mathbb{F}_q^{m \times n}$ is {$(t-1)$-intersection-free} if $\dim \ker(A-B) \neq t-1$ for all $A,B \in \mathcal{F}$. A \emph{forbidden $(t-1)$-intersection problem} for a collection of matrices asks for the size and structure of extremal $(t-1)$-intersection-free families within that collection. We solve this problem in $\mathrm{GL}(n,q)$ for all pairs $(n,t)$ such that $t<c\cdot n$ where $c$ is a universal constant. We show that the $t$-umvirates and their duals, are the only maximal $(t-1)$-intersection-free families $\mathcal{F} \subset \mathrm{GL}(n,q)$. Here, a $t$-umvirate is defined as the family of all matrices that agree on a fixed $t$-dimensional subspace, and its dual as those whose transposes agree on it. The best previously known result, due to Ellis, Kindler, and Lifshitz, established this bound under the assumption $n \geq e^{Ct\log t}$ for some constant $C>0$. We also give Frankl--Rödl-type constructions showing that this range of $t$ is almost the best possible: we show that for values of $t>n/2$ the extremal behavior changes and no clean analogue is expected. Our proof builds upon recent global hypercontractivity results for matrix spaces due to Evra, Kindler, and Lifshitz, and broadly applies to any sufficiently dense class of matrices.

In a recent paper, the authors introduced an infinite hierarchy of $q$-hypergeometric identities, of which the first three orders, $0$, $1$, and $2$, relate to the partition theorems of Euler, Lebesgue, and Capparelli, and stated a partition theorem at order 4 which lies beyond Capparelli's theorem. Here, we first state certain partition theorems that hold at all even orders beyond Capparelli and provide bijective proofs for these theorems. In doing so, we show that there is a fourfold infinite hierarchy of partition theorems that emanates from Capparelli's theorem, which is the base case. It is also shown that the equality of two of the four generating functions holds for all orders, odd and even. Lastly, a very general framework for the remaining two functions is constructed via the method of weighted words, encompassing all possible orders and yielding several infinite hierarchies with different dilations and translations.

In 1988, Las Vergnas conjectured that if $M$ is a binary matroid with bicycle dimension $d$, then for $0 \leq k \leq d$, the $k$th derivative of the diagonal Tutte polynomial $T(M;z,z)$ evaluated at $z=-1$ is an integer multiple of $2^{d-k}$. While this was rapidly disproved for binary matroids and for graphs in general, extensive computations strongly suggested that it might be true for planar graphs. In this paper we prove that this is indeed the case. To do this, we consider a stronger divisibility property that we call the LV property, and a larger class of graphs, namely the class of delta-wye-reducible graphs. By a detailed analysis of how a delta-wye exchange affects the coefficients of the diagonal Tutte polynomial, we show that delta-wye-reducible graphs have the LV property. That Las Vergnas' conjecture holds for planar graphs immediately follows because planar graphs are delta-wye reducible and the LV property is stronger than Las Vergnas' divisibility conditions.

A cyclic meander is an embedded oriented loop in the plane intersecting a fixed infinite line, or circle, transversely in a linearly ordered set of $2n$ points. By keeping track of the order in which the loop visits these points, the cyclic meander induces a cyclic permutation on these marked points. Correspondingly, given a permutation on $n$ letters, one can ask whether or not a cyclic meander induces the permutation in this manner, and if not, what is the most efficient way of doing so if we allow more points of intersection? This process gives a way of associating to a permutation on $n$ letters a measurement of complexity of the permutation in question. The principal result of this work shows that the maximum of this quantity, the \emph{meander number}, over all cyclic permutations on $n$ letters, is bounded above and below quadratically in $n$. This result resolves a conjecture of Schwartz~\cite{richtpss} in relation to his work on the topological salesman problem. We conclude this work by constructing families of cyclic permutations on $n$ letters whose meander numbers realize a continuum of growth rates between linear and quadratic.

The cooling number measures the speed at which a slow-moving influence or contagion spreads on a graph. In this paper, we investigate the cooling number of four classical graph products: the Cartesian product, the strong product, the lexicographic product, and the direct product. We also determine the cooling number of a disconnected graph in terms of the cooling numbers of its components. We conclude with open problems.

In arXiv:2311.10069, Matolcsi et al. show that the fractional chromatic number of the plane is at least 4. Their proof uses a 27-vertex unit-distance graph in the Moser lattice, with geometric fractional chromatic number exactly 4. We show that this bound is tight for graphs in the Moser lattice by exhibiting geometric 4-colorings of the entire lattice. The same colorings also extend to the entire Moser ring.

Let $G$ be a loopless multigraph with maximum degree $\Delta(G)$, average degree $\overline{d}(G)$, density $\Gamma(G)$, and chromatic index $\chi'(G)$. A multigraph $G$ is called edge-$\Delta$-critical if $\Delta(G)=\Delta$, $\chi'(G)=\Delta(G)+1$ and $\chi'(H) \le \Delta(G)$ for every proper subgraph $H\subset G$. Vizing conjectured that if $G$ is an edge-$\Delta$-critical simple graph on $n$ vertices, then $\overline{d}(G) \ge \Delta-1+\tfrac{3}{n}$. Motivated by this, we conjecture that every edge-$\Delta$-critical multigraph $G$ satisfies $\overline{d}(G) \ge \tfrac{2\Delta+2}{3}$, which is best possible. We first give a general lower bound in this direction. For any such graph $G$, \[ \overline{d}(G) \ge \begin{cases} \frac{\sqrt{17}-3}{2}(\Delta+1) & \text{if } \Delta \le 112;\\[4pt] \frac{\Delta+\sqrt{2\Delta-1}}{2} & \text{if } \Delta \ge 113. \end{cases} \] This bound can be further improved under an additional condition on the multiplicity $\mu$. In this case, \[ \overline{d}(G)\ge \min\left\{ \frac{2\mu\Delta+2\mu(2\mu-1)}{4\mu-1},\; \frac{\sqrt{17}-3}{2}(\Delta+1) \right\}. \] We also confirm the conjecture for $\Delta \in \{2,3,4,5,6,7,8\}$. As a consequence, Goldberg's conjecture~\cite{Goldberg1984} holds for $\Delta(G)\in\{2,3,4,5\}$, that is, every multigraph $G$ with $\chi'(G)\ge \Delta(G)+1$ satisfies $\Gamma(G)\ge \Delta(G)$.

We prove a prescribed-leftover-chord theorem for Hamiltonian Berge cycles of odd order. Let $C$ be a Hamiltonian Berge cycle on $n=2r+1$ vertices, and let $\mathcal G$ be a set of hyperedges, all of size at least $r$, containing the hyperedges of $C$. If $D\subseteq\{2,\ldots,r\}$ and $|\mathcal G|\ge n+|D|$, then the hyperedges can be reassigned to the adjacent pairs of the same cyclic order so that, for each $d\in D$, a distinct unused hyperedge realizes cyclic distance $d$. Consequently, the odd-order case of the one-extra-edge question of Bailey, Hollars, Li and Luo has an affirmative answer for all $n=2r+1\ge7$, in the convention including Berge cycles of length $2$. The proof combines an additive lemma in $\mathbb Z_{2r+1}$ with an alternating matching exchange.

Brouwer's Laplacian conjecture states that the sum of the largest $k$ eigenvalues of a graph's Laplacian is less than or equal to the number of edges plus $\binom{k+1}{2}$. We give a proof of this conjecture. Our proof relies on the Grone--Merris--Bai theorem for \emph{split} graphs. We also show the converse, thereby establishing an equivalence between Brouwer's conjecture and the Grone--Merris--Bai theorem.

We show that for any $A\subseteq \mathbb{F}_q^n$ lacking non-trivial solutions to a translation-invariant linear equation of genus one, meaning that no nonempty proper subset of the coefficients sums to $0$, there is a set $B\subseteq \mathbb{F}_q^m$ in some higher dimension which also lacks non-trivial solutions, such that \[|B|^{1/m}>|A|^{1/n}.\] In particular, this implies that no fixed cap set in $\mathbb{F}_3^n$ gives an asymptotically optimal lower bound by direct products alone.

Given a finite group $G$, we prove that there exist signs $\varepsilon\in\{\pm1\}^G$ such that $$\left\| \sum_{g\in G} \varepsilon_g\rho(g) \right\|\leq C\, \sqrt{|G|},$$ where $\rho$ is the left regular representation of $G$, and $C$ is a universal constant. This special case of the Matrix Spencer conjecture was posed in [BKMZ24], where it was established for simple groups.

Let $M \subset \{0,\pm1\}^n$ be a set such that $(m,m)=4$ for every $m\in M$, and $(m_1,m_2)\in\{-4,-3,-2,-1,0,3\}$ for any two distinct vectors $m_1,m_2\in M$. We determine the maximum possible cardinality of such a set $M$ for all sufficiently large $n$.

Let $S$ and $T$ be disjoint sets with $|S|=i$ and $|T|=r-1$ for $2\le i\le r-1$, and let $B_i^{(r)}$ be the $r$-graph on $S\cup T$ whose edges are the $r$-subsets containing $S$ or $T$. We study the deficit $q_{r,i}:=1-\pi(B_i^{(r)})$ in its Turán density. Balogh, Bohman, Bollobás, and Zhao previously obtained bounds for these deficits with logarithmic gaps near both ends of the sequence $B_i^{(r)}$, namely, when $i=O(1)$ or $i=r-O(1)$. We close these gaps by showing that, as $r\to\infty$, for every fixed integer $a\ge1$, $q_{r,a+1}=\Theta_a(r^{-a})$, and for every fixed integer $b\ge2$, $q_{r,r-b}=\Theta_b(r^{-b}\log r)$.

For a given Coxeter group, we study distributive lattices called middle orders refining the weak order and refined by the Bruhat order. In type $A$, we construct such lattices indexed by binary trees using a direct bijection between permutations and lower sets of a certain partition of the root poset into rectangles. When the binary tree is a left-comb tree, we recover the middle order defined by Bouvel, Ferrari, and Tenner (2025). We study combinatorial properties of these lattices, and show they are the only distributive lattices between the weak and Bruhat orders in type $A$. For general Coxeter groups, we study middle orders on parabolic quotients and use these to generalize our construction in type $A$ to other Weyl groups, obtaining so-called ``minuscule middle orders''. We show that they are a subset of sorting orders defined by Armstrong (2009), and we give conjectural descriptions of all middle orders that are not minuscule.

For integers $k\ge 1$ and $n\ge k+1$, let $\operatorname{ex}^{\mathrm{HIST}}_k(n)$ denote the maximum number of edges in a $k$-connected graph of order $n$ which contains no homeomorphically irreducible spanning tree (or briefly HIST). We determine these extremal numbers for $k=1$ and $k=2$. More precisely, we prove that $\operatorname{ex}^{\mathrm{HIST}}_1(n)=\binom{n-2}{2}+2$ for $n\ge 9$, with $L_n$ as the unique extremal graph, and that $\operatorname{ex}^{\mathrm{HIST}}_2(n)=\binom{n-3}{2}+4$ for $n\ge 13$, with $B_n$ as the unique extremal graph. This provides a Turán-type extremal result for spanning trees with no vertices of degree two.

We develop a framework for comparing probability measures on finite posets via hockey-stick domination, an order relation defined through interval-counting test functions. The theory introduces poset integrals, derivatives, power functions and the associated moment functionals, all of which are invariant under poset isomorphisms. We prove that hockey-stick domination admits an exact quantitative characterization: whenever $\mu$ is dominated by $\nu$ in the hockey-stick order, the corresponding Zolotarev-type distance is equal to one half of the second-order poset moment of $\nu-\mu$. We further develop a constructive theory for generating such domination relations. In particular, we show that hockey-stick domination is preserved under direct products, disjoint unions, ordinal sums, and suitable ideal restrictions, yielding natural families of examples on chains, Boolean posets, rectangular lattices, rooted trees, and Young diagrams.

Let $\varepsilon\in (0,1/2]$ and let $G$ be a balanced bipartite graph on $2n$ vertices with minimum degree at least $(1/2 + \varepsilon)n$. Then, whp, the hitting time for minimum degree 2 coincides with the hitting time for Hamiltonicity. This extends Bollobás--Kohayakawa and gives a bipartite analogue of Johansson's theorem. As an immediate corollary, we deduce a sharp threshold result for Hamiltonicity in such graphs.

Lifted product codes are an important family of quantum low-density parity-check (QLDPC) codes, as they were the first QLDPC code family shown to be asymptotically good. Understanding the structure of their parity-check matrices $H_{\mathsf{X}}$ and $H_{\mathsf{Z}}$, as well as the associated Tanner graphs, is essential for analyzing their decoding behavior and error-floor performance. In this work, we show that the Tanner graphs of $H_{\mathsf{X}}$ and $H_{\mathsf{Z}}$ are indeed isomorphic, and investigate their graph-theoretical structure. We establish conditions ensuring the connectivity of these graphs and provide bounds on their minimal absorbing sets, providing new insight into the combinatorial structures influencing decoding performance.

We describe a computational search for quadratic APN (Almost Perfect Nonlinear) functions in dimension 8 within a structured self-equivalence subspace. The search space is a 40-dimensional binary linear subspace consisting of all functions commuting with a linear automorphism of order 5 (class 22 in the taxonomy of Beierle, Brinkmann, and Leander, 2021), previously reported to contain no APN functions. Our approach combines random sampling via an explicit RREF parameterization (approximately 600 fresh APN-positive evaluations per core-hour) with Gröbner basis computation in Magma to enumerate all APN functions in a 24-dimensional hyperplane through each center (approximately 10 minutes per hyperplane). From 428 hyperplane computations, covering 0.65% of all 65,536 hyperplanes, we obtained 566 quadratic APN functions forming six CCZ-equivalence classes under the ortho-derivative invariant. Four classes, comprising 500 functions, match no entry in the 2025 database of 3,775,599 quadratic APN functions or in the pre-2020 compilation of 12,921 instances. Two classes (66 functions) are CCZ-equivalent to the Gold functions x^3 and x^9, confirming the correctness of the search pipeline. A membership analysis shows that the three new classes (B, C, D) lie entirely outside the original subspace and occur only in Gold-centered slices, demonstrating the essential role of the Gröbner basis stage. In 532 experiments using database functions as slice centers and 20 experiments with random centers, no APN neighbors were found, indicating that the gateway phenomenon is specific to the self-equivalence structure of the search space. Since the ortho-derivative invariant is a complete CCZ-invariant for quadratic APN functions, the absence of matching signatures provides a rigorous proof of CCZ-inequivalence.

For a set $X$ of integer points, the relaxation complexity $\operatorname{rc}(X)$ is the smallest number of facets of any polyhedron $P$ such that $P \cap \mathbb{Z}^d = X$. In this paper, we focus on the case where $X$ is the discrete standard simplex $\Delta_d = \{\mathbf{0}, \mathbf{e}_1, \dots, \mathbf{e}_d\}$. We show that $\operatorname{rc}(\Delta_d) = O(\log d)$ by an explicit, elementary construction. This improves upon the previously best-known upper bound $\operatorname{rc}(\Delta_d) = O(d / \sqrt{\log d})$ due to Aprile, Averkov, Di Summa, and Hojny (2024) and matches an asymptotic lower bound by Averkov and Schymura (2022).

Let $L(\lambda)$ be a simple highest weight module of a classical Lie algebra $\mathfrak{g}$ with highest weight $\lambda-\rho$, where $\rho$ is half the sum of positive roots. Joseph proved that the associated variety of the annihilator ideal of $L(\lambda)$ (also called the annihilator variety) is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. Recently, Bai--Ma--Wang introduced a partition algorithm to describe this corresponding nilpotent orbit for a given highest weight module $L(\lambda)$. In this paper, we present a new direct proof of Bai--Ma--Wang's partition algorithm using Sommers duality.

In this work, we settle a problem that dates back to the early 2000s. We provide a complete characterization of special matchings of arbitrary Bruhat intervals in Coxeter groups of type $A$ and apply this result to prove a conjecture of Brenti from 2003 concerning the computation of Kazhdan-Lusztig $R$-polynomials via special matchings. This yields new evidence in support of the Combinatorial Invariance Conjecture.

The process of deriving the Desarguesian plane $PG(2,q^2)$ to get the Hall plane is well known, and the problem of when a conic in $PG(2,q^2)$ inherits to an arc in the Hall plane has been solved. In this article we look at the generalisation of replacing an André net of $PG(2,q^t)$, $t\geq 3$ to construct an André plane of order $q^t$. This article looks at the case where $q$ is odd and $t$ is prime, and determines when a conic in $PG(2,q^t)$ inherits to an arc in an André plane. Further, the number of arcs in an André plane that are inherited in this way is enumerated.

A graph $G$ is $k$-$colorable$ if $V(G)$ can be partitioned into at most $k$ stable sets. A graph $G$ is $k$-$chromatic$ if $k$ is the smallest integer for which $G$ is $k$-colorable. In general, for a fixed $k\ge 3$, determining whether an arbitrary graph $G$ is $k$-colorable is NP-complete. Consequently, $k$-coloring algorithms for restricted graph classes, such as $\mathcal{H}$-free graphs, have been widely studied over the past few decades. A graph $G$ is $k$-$vertex$-$critical$ if $G$ is $k$-chromatic and every proper induced subgraph of $G$ is ($k$-1)-colorable. Given a graph $G$, most of the certifying $k$-coloring algorithms in the literature either output a $k$-coloring of $G$ or a ($k$+1)-vertex-critical induced subgraph of $G$, thus, proving that $G$ is not $k$-colorable. As a result, $k$-vertex-critical graphs have gathered considerable attention in the recent years. Beaton and Cameron [Vertex-critical graphs in co-gem-free graphs, Theoretical Computer Science 1042 (2025) 115234] asked for which graphs $H$ of order five are there finitely many $k$-vertex-critical (co-gem, $H$)-free graphs for all $k$? In this paper we explore the structure of (co-gem, house)-free graphs and (co-gem, dart)-free graphs, and prove that, for each $k\ge 1$, there are finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, when $H$ is in $\{$house, dart$\}$.

We study discrete analogues of classical spectral geometric inequalities and extremal eigenvalue problems on graphs. The well-known Krahn--Szegő inequality states that the minimum of $\lambda_2(\Omega)$ among bounded open sets of $\mathbb{R}^n$ with given volume is achieved by the union of two identical balls $\mathbb{R}^n$. Firstly, we establish a Krahn--Szegő type inequality for trees. For trees with a fixed number of interior vertices and boundary leaves, we completely characterize the extremal structures that minimize the second Dirichlet eigenvalue. Secondly, we develop a nodal domain method for adjacency matrices. By proving a nodal domain theorem in adjacency version for graphs, we obtain upper bounds for the second largest adjacency eigenvalue $\rho_2(G)$ of $G$ in given graph classes. These bounds imply some previous results. Finally, we settle the Aouchiche--Hansen conjecture (2010) on the second largest eigenvalue with given number of edges and clique number. We prove that for connected graphs $G$ of odd order $n \geq 5$, $|\rho_2| \cdot \omega \leq m-2$, with equality if and only if $G$ consists of two complete graphs of orders $\frac{n+1}{2}$ and $\frac{n-1}{2}$ joined by an edge or a path. For even $n \geq 2$, the quantity $|\rho_2| \cdot \omega - m$ is maximized exactly when $G$ is the join of two copies of $K_{n/2}$ by an edge. The core of the methods developed in this paper is to regard a connected graph as an internally disconnected graph with Dirichlet boundary condition. This perspective allows us to transfer nodal domain techniques from continuous spectral geometry to discrete settings and to obtain sharp extremal characterizations across diverse graph classes.

When $S$ is a finite set of positive integers, we can consider classical Subtraction Nim with $S$ as the set of removable numbers. Even when $S$ consists of three elements, many questions remain unanswered. For example, we do not have a period formula of the Nim value. In this paper, we generalize $S$ to be a finite set of positive real numbers. We found that in some regions, we can give concrete formulae for the period and the Nim value function. In particular when $S$ consists of three elements, we found sufficient conditions for the Nim value function to be purely periodic with the period which is equal to the sum of two of elements of $S$. To be more precise, let $S = {a,b,c}$ with $0 < a < b < c$, then for example when $a \leq b \leq 2a$ with $a+b \geq c$, the Nim value function is purely periodic with a period $a+c$. There are much more regions with precise period formulae. We have also some generalizations for the cases $|S| \geq 4$. Even when $S$ consists of integers, these results seem to be new.

For every fixed $t\ge 1$, we prove a parity analogue of the mighty Erdős-Hajnal property for $t$-intersecting curves in the plane. Let $\mathcal B$ be a set of blue curves and $\mathcal G$ a set of green curves in the plane such that $\mathcal B\cup\mathcal G$ is a collection of $t$-intersecting curves in general position. We show that there exist subfamilies $\mathcal B'\subseteq\mathcal B$ and $\mathcal G'\subseteq\mathcal G$ such that $|\mathcal B'|\geq \varepsilon|\mathcal B|$ and $|\mathcal G'|\geq \varepsilon|\mathcal G|$, where $\varepsilon>0$ depends only on $t$, such that either every pair in $\mathcal B'\times\mathcal G'$ intersects an even number of times or every such pair intersects an odd number of times. For $t=1$, this recovers the theorem of Fox, Pach, and Suk for pseudo-segments. As an application, we show that every $n$-vertex topological graph with edges forming a $t$-intersecting family and with no $k$ edges that pairwise cross an odd number of times has at most $n(\log n)^{O_t(\log k)}$ edges.

Let $\lambda_i(G)$ denote the $i$-th largest eigenvalue of adjacency matrix of a graph $G$. Gerschgorin's Theorem indicates $\lambda_1(G)$ belongs to the largest disk, i.e., $\lambda_1(G)\le\Delta_1(G)$, where $\Delta_i(G)$ is the $i$-th largest degree of $G$. We show that $\lambda_2(G)$ lies in the second largest disk. That is, in detail, $$\lambda_2(G)<\Delta_2(G)-\frac{1}{n^2}.$$ A classical theorem proved by Hong [\textit{Linear Algebra Appl.} 1988] states that $\lambda_1(G)\le\sqrt{2m-n+1}$ for a connected graph $G$ with $n$ vertices and $m$ edges, where the equality holds if and only if $G$ is a star $S_n$ or a complete graph $K_n$. We give a refinement of Hong's theorem by showing $$\lambda_1(G)<\sqrt{2m-n}$$ for any connected graph $G\not\in\left\{S_n,S^1_{n-1},K_n,K^1_{n-1}\right\}$. Based on this improved upper bound of $\lambda_1(G)$, for a connected graph $G$ with $n$ vertices and $m$ edges, we are able to prove a sharp upper bound of $\lambda_2(G)$ that $$\lambda_2(G)\le\sqrt{m-\frac{n}{2}-\frac{1}{2}},$$ except $G$ is obtained from two disjoint $S_\frac{n}{2}$ by adding an edge between a pendant vertex of each star. Moreover, we provide a complete characterization to extremal graphs attaining the equality.

In this paper, we demonstrate the effects on the second largest eigenvalue $\lambda_2(G)$ of a connected graph $G$ after edge addition or deletion. In 1989, Chung, Graham and Wilson showed $\max\{|\lambda_2|,|\lambda_n|\}>\Omega(n)$ for dense $K_{r+1}$-free graphs of order $n$, giving spectral comprehension of existence of large clique or independent set, respect to Ramsey theory. Applying the results of effects on $\lambda_2$ after edge operations, we determine the maximum value of $\lambda_2$ among all $K_{r+1}$-free connected graphs with given order, and completely characterize the extremal graphs. Moreover, for arbitrary given graph $F$, we investigates the maximum second largest $\lambda_2(G)$ among $F$-free connected graphs of order $n$. Let $\rho^*(n,F)$ be the maximum spectral radius of $F$-free graphs on $n\ge n_F$ vertices, and $G^*(n,F)$ be a graph with its spectral radius $\rho\big(G^*(n,F)\big)=\rho^*(n,F)$. We prove that, for an $F$-free connected graph $G$ of order $n\ge f(n_F)$, \\(1) if $n$ is odd, then $$\lambda_2(G)\le\rho^*\left(\frac{n-1}{2},F\right)$$ with equality if and only if $G\in \mathcal{I}\big(G^*(\frac{n-1}{2},F),G^*(\frac{n-1}{2},F)\big)$; and\\ (2) if $n$ is even, and $F$ does not contain cut edges, then the graph $G^†$ with the maximum second largest eigenvalue satisfies $$\lambda_2(G^†)=\rho^*\left(\frac{n}{2},F\right)-o(1)$$ and $G^†\in \mathcal{E}\big(H_1,H_2\big)$, where $H_1$ and $H_2$ are $F$-saturated graphs on $\frac{n}{2}$ vertices. In particular, other than a complete graph $K_{r+1}$, when $F$ is a book graph $B_{k+1}$ or an odd cycle $C_{2k+1}$, we are able to determine the maximum second largest eigenvalue for $F$-free connected graphs of given order, and completely characterize the extremal graphs.