The Merino-Welsh conjecture states that for a graph G without loops and bridges the Tutte polynomial TG(x, y) satisfies the inequality max(TG(2, 0), TG(0, 2)) TG(1,1). Later Jackson proved that for any matroid M without loops and coloops we have TM(3, 0)TM(0, 3) TM(1, 1)2. The value 3 in this statement was improved to 2.9243 by Beke, Cs & aacute;ji, Csikv & aacute;ri and Pituk. In this paper, we further improve on this result by showing that TM(2.355, 0)TM(0, 2.355) TM(1, 1)2. We also prove that the Merino-Welsh conjecture is true for matroids M, where all circuits of M and its dual M* have length between & ell; and (& ell; - 2)2(& ell;2-4 & ell; + 2) for some & ell; 4. (c) 2026 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Given a connected graph, the principal eigenvector of the adjacency matrix (often called the Perron vector) can be used to assign positive weights to the vertices. A natural way to measure the homogeneousness of this vector is by considering the ratio of its & ell;(1) and & ell;(2) norms. It is easy to see that the most balanced graphs in this sense (i.e., the ones with the largest ratio) are the regular graphs. What can we say about the least balanced (or most centralized) graphs with the smallest ratio? It was conjectured by R & uuml;cker, R & uuml;cker and Gutman that, for any given n >= 6, among n-vertex connected graphs the smallest ratio is achieved by the complete graph K-4 with a single path Pn-4 attached to one of its vertices. In this paper we confirm this conjecture. We also verify the analogous conjecture for trees: for any given n >= 8, among n-vertex trees the smallest ratio is achieved by the star graph S5 with a path Pn-5 attached to its central vertex. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license

In this paper we study the following extremal graph theoretic problem: Given an undirected Eulerian graph G, which Eulerian orientation minimizes or maximizes the number of arborescences? We solve the minimization for the complete graph K-n, the complete bipartite graph K-n,K-m, and for the so-called double graphs, where there are even number of edges between any pair of vertices. In fact, for K-n we prove the following stronger statement. If T is a tournament on n vertices with out-degree sequence d(1)(+), ... , d(n)(+), then 1 allarb(T ) >= 1/n(Pi(n)(k=1) (d(k)(+) + 1) + Pi(n)(k=1) d(k)(+), where allarb(T) is the total number of arborescences. Equality holds if and only if T is a locally transitive tournament. We also give an upper bound for the number of arborescences of an Eulerian orientation for an arbitrary graph G. This upper bound can be achieved on Kn for infinitely many n.

The matroidal version of the Merino–Welsh conjecture states that the Tutte polynomial TM(x,y) of any matroid M without loops and coloops satisfies thatmax⁡(TM(2,0),TM(0,2))⩾TM(1,1). Equivalently, if the Merino–Welsh conjecture is true for all matroids without loops and coloops, then the following inequalities are also satisfied for all matroids without loops and coloops:TM(2,0)+TM(0,2)⩾2TM(1,1), andTM(2,0)TM(0,2)⩾TM(1,1)2. We show a counter-example for these inequalities.

The classical Tutte polynomial is a two-variate polynomial TG(x,y) associated to graphs or more generally, matroids. In this paper, we introduce a polynomial T˜H(x,y) associated to a bipartite graph H that we call the permutation Tutte polynomial of the graph H. It turns out that TG(x,y) and T˜H(x,y) share many properties, and the permutation Tutte polynomial serves as a tool to study the classical Tutte polynomial. We discuss the analogues of Brylawsi’s identities and Conde–Merino–Welsh type inequalities. In particular, we will show that if H does not contain isolated vertices, then T˜H(3,0)T˜H(0,3)≥T˜H(1,1)2,which gives a short proof of the analogous result of Jackson: TG(3,0)TG(0,3)≥TG(1,1)2 for graphs without loops and bridges. We also give improvement on the constant 3 in this statement by showing that one can replace it with 2.9243.

For a graph G let ε(G) denote the number of Eulerian orientations, and v(G) denote the number of vertices of G. We show that if (G_n)_n is a sequence of Eulerian graphs that are convergent in Benjamini–Schramm sense, then lim_n→∞1/v(G_n)lnε(G_n) is convergent.

The spectral radius of a graph is the spectral radius of its adjacency matrix. A threshold graph is a simple graph whose vertices can be ordered as v_1, v_2, …, v_n, so that for each 2 ≤ i ≤ n, vertex v_i is either adjacent or nonadjacent simultaneously to all of v_1, v_2, …, v_i-1. Brualdi and Hoffman initially posed and then partially solved the extremal problem of finding the simple graphs with a given number of edges that have the maximum spectral radius. This problem was subsequently completely resolved by Rowlinson. Here, we deal with the similar problem of maximizing the spectral radius over the set of connected simple graphs with a given number of vertices and edges. As shown by Brualdi and Solheid, each such extremal graph is necessarily a threshold graph. We investigate the spectral radii of threshold graphs by relying on computations involving lazy walks. Furthermore, we obtain three lower bounds and one upper bound on the spectral radius of a given connected threshold graph.

In this short note we show that a system M = (E, r) with a ground set E of size m and (rank) function r : 2E -> Z >= 0 satisfying r(S) <= min(r(E), |S|) for every set S subset of E, the Tutte polynomial n-ary sumation TM(x, y) := S subset of E (x-1)r(E)-r(S)(y -1)|S|-r(S), written as TM(x, y) = n-ary sumation i,j tijxiyj, satisfies that for any integer h >= 0, we have n-ary sumation h i=0 h-i n-ary sumation j=0 (h- i)(h-r) (-1)jtij= (-1)m-r , j h - m where r = r(E), and we use the convention that when h < m, the binomial coefficient ( h-r ) is interpreted as 0. h-m This generalizes a theorem of Brylawski on matroid rank functions and h < m, and a theorem of Gordon for h <= m with the same assumptions on the rank function. The proof presented here is significantly shorter than the previous ones. We only use the fact that the Tutte polynomial TM(x, y) simplifies to (x-1)r(E)y|E| along the hyperbola (x-1)(y- 1) =1. (c) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

We show that if G is a d–regular graph on n vertices, then the number of spanning forests F ( G ) satisfies F ( G ) ≤ d n. The previous best bound due to Kahale and Schulman gave ( d + 1 / 2 + O ( 1 / d ) ) n. We also have the more precise conjecture that F ( G ) 1 / n ≤ ( d − 1 ) d − 1 ( d 2 − 2 d − 1 ) d / 2 − 1 . If this conjecture is true, then the expression on the right hand side is the best possible.

We study a class of determinant inequalities that are closely related to Sidorenko's famous conjecture (also conjectured by Erdos and Simonovits in a different form). Our main result can also be interpreted as an entropy inequality for Gaussian Markov random fields (GMRF). We call a GMRF on a finite graph G homogeneous if the marginal distributions on the edges are all identical. We show that if G is bipartite, then the differential entropy of any homogeneous GMRF on G is at least vertical bar E(G)vertical bar times the edge entropy plus vertical bar V(G)vertical bar-2 vertical bar E(G)vertical bar times the point entropy. We also show that in the case of non-negative correlation on edges, the result holds for an arbitrary graph G. The connection between Sidorenko's conjecture and GMRF's is established via a large deviation principle on high dimensional spheres combined with graph limit theory. It is also observed that the system we study exhibits a phase transition on large girth regular graphs. Connection with Ihara zeta function and the number of spanning trees is also discussed.

Graph covers and the Bethe free energy (BFE) have been useful theoretical tools for producing lower bounds on a variety of counting problems in graphical models, including the permanent and the ferromagnetic Ising model. Here, we investigate weighted homomorphism counting problems over bipartite graphs that are related to a conjecture of Sidorenko. We show that the BFE does yield a lower bound in a variety of natural settings, and when it does yield a lower bound, it necessarily improves upon the lower bound conjectured by Sidorenko. Conversely, we show that there exist bipartite graphs for which the BFE does not yield a lower bound on the homomorphism number. Finally, we use the characterizations developed as part of this work to provide a simple proof of Sidorenko’s conjecture in a number of special cases.

For a graph G=(V,E) with v(G) vertices the partition function of the random cluster model is defined by Z_G(q,w)=∑ _A⊆ E(G)q^k(A)w^|A|, where k(A) denotes the number of connected components of the graph (V, A). Furthermore, let g(G) denote the girth of the graph G, that is, the length of the shortest cycle. In this paper we show that if (G_n)_n is a sequence of d-regular graphs such that the girth g(G_n)→∞ , then the limit lim _n→∞1/v(G_n)ln Z_G_n(q,w)=lnΦ _d,q,w exists if q≥ 2 and w≥ 0 . The quantity Φ _d,q,w can be computed as follows. Let Φ _d,q,w(t):= ( √(1+w/q)cos (t)+√((q-1)w/q)sin (t)) ^d + (q-1)( √(1+w/q)cos (t) -√(w/q(q-1))sin (t)) ^d, then Φ _d,q,w:=max _t∈ [-π ,π ]Φ _d,q,w(t), The same conclusion holds true for a sequence of random d-regular graphs with probability one. Our result extends the work of Dembo, Montanari, Sly and Sun for the Potts model (integer q), and we prove a conjecture of Helmuth, Jenssen and Perkins about the phase transition of the random cluster model with fixed q.

Let T G ( x , y ) be the Tutte polynomial of a graph G . In this paper we show that if ( G n ) n is a sequence of d -regular graphs with girth g ( G n ) → ∞, then for x ≥ 1 and 0 ≤ y ≤ 1 we have lim n → ∞ ⁡ T G n ( x , y ) 1 / v ( G n ) = t d ( x , y ) , where t d ( x , y ) = { ( d − 1 ) ( ( d − 1 ) 2 ( d − 1 ) 2 − x ) d / 2 − 1 if x ≤ d − 1 and 0 ≤ y ≤ 1 , x ( 1 + 1 x − 1 ) d / 2 − 1 if x > d − 1 and 0 ≤ y ≤ 1 . If ( G n ) n is a sequence of random d -regular graphs, then the same statement holds true asymptotically almost surely. This theorem generalizes results of McKay (x = 1 , y = 1, spanning trees of random d -regular graphs) and Lyons (x = 1 , y = 1, spanning trees of large-girth d -regular graphs). Interesting special cases are T G ( 2 , 1 ) counting the number of spanning forests, and T G ( 2 , 0 ) counting the number of acyclic orientations.

Let G be a triangle-free graph on n vertices with adjacency matrix eigenvalues μ1(G)≥μ2(G)≥…≥μn(G). In this paper we study the quantityμ1(G)+μn(G). We prove that for any triangle-free graph G we haveμ1(G)+μn(G)≤(3−22)n. This was proved for regular graphs by Brandt, we show that the condition on regularity is not necessary. We also prove that among triangle-free strongly regular graphs the Higman-Sims graph achieves the maximum ofμ1(G)+μn(G)n.

For a graph $G$ on $v(G)$ vertices let $m_k(G)$ denote the number of matchings of size $k$, and consider the partition function $M_{G}(\lambda)=\sum_{k=0}^nm_k(G)\lambda^k$. In this paper we show that if $G$ is a $d$--regular graph and $0 \frac{1}{v(K_{d+1})}\ln M_{K_{d+1}}(\lambda).$$ The same inequality holds true if $d=3$ and $\lambda<0.3575$. More precise conjectures are also given.

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have absolute value at most $4$. We more generally show that roots of the edge cover polynomial of a $k$-uniform hypergraph have absolute value at most $2^k$, and discuss applications of this to the roots of domination polynomials of graphs. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.

Given a graph $G$ with only even degrees, let $\varepsilon(G)$ denote the number of Eulerian orientations, and let $h(G)$ denote the number of half graphs, that is, subgraphs $F$ such that $d_F(v)=d_G(v)/2$ for each vertex $v$. Recently, Borbényi and Csikvári proved that $\varepsilon(G)\geq h(G)$ holds true for all Eulerian graphs, with equality if and only if $G$ is bipartite. In this paper we give a simple new proof of this fact, and we give identities and inequalities for the number of Eulerian orientations and half graphs of a $2$-cover of a graph $G$.

The goal of this paper is to advertise the method of gauge transformations (aka holographic reduction, reparametrization) that is well-known in statistical physics and computer science, but less known in combinatorics. As an application of it we give a new proof of a theorem of A. Schrijver asserting that the number of Eulerian orientations of a d–regular graph on n vertices with even d is at least dd∕22d∕2n. We also show that a d–regular graph with even d has always at least as many Eulerian orientations as (d∕2)–regular subgraphs.

Graph covers and the Bethe free energy have been useful theoretical tools for producing lower bounds on a variety of counting problems in graphical models, including the permanent and the ferromagnetic Ising model. Here, we propose a new conjecture that the Bethe free energy yields a lower bound on the weighted homomorphism counting problem over bipartite graphs. We show that this conjecture strengthens existing conjectures, and we prove the conjecture in several special cases using a novel reformulation of the graph cover characterization of the Bethe free energy.

In this paper, we survey some recent developments on statistical properties of matchings of very large and infinite graphs. We discuss extremal graph theoretic results like Schrijver’s theorem on the number of perfect matchings of regular bipartite graphs and its variants from the point of view of graph limit theory. We also study the number of matchings of finite and infinite vertex-transitive graphs.