Transactions on Combinatorics
Volume:10 Issue: 2, Jun 2021

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 $0<lambda<(4d)^{-2}$‎, ‎then‎ ‎$$frac{1}{v(G)}ln M_G(lambda)>frac{1}{v(K_{d+1})}ln M_{K_{d+1}}(lambda).$$‎ ‎The same inequality holds true if $d=3$ and $lambda<0.3575$‎. ‎More precise conjectures are also given‎.

‎Cruz‎, ‎Monsalve and Rada [Extremal values of vertex-degree-based topological indices of chemical trees‎, ‎Appl‎. ‎Math‎. ‎Comput‎. ‎380 (2020) 125281] posed an open problem to find the maximum value of the exponential second Zagreb index for chemical trees of given order‎. ‎In this paper‎, ‎we solve the open problem completely‎.

‎The characterization of the ideal access structures is one of the main open problems in secret sharing and is important from both practical and theoretical points of views‎. ‎A graph-based $3-$homogeneous access structure is an access structure in which the participants are the vertices of a connected graph and every subset of the vertices is a minimal qualified subset if it has three vertices and induces a connected graph‎. ‎In this paper‎, ‎we introduce the graph-based $3-$homogeneous access structures and characterize the ideal graph-based $3$-homogeneous access structures‎. ‎We prove that for every non-ideal graph-based $3$-homogeneous access structure over the graph $G$ with the maximum degree $d$ there exists a secret sharing scheme with an information rate $frac{1}{d+1}$‎. ‎Furthermore‎, ‎we mention three forbidden configurations that are useful in characterizing other families of ideal access structures‎.

In this paper, we compute a formula for the $a$-number of certain maximal curves given by the equation $y^{q}+y=x^{frac{q+1}{2}}$ over the finite field $mathbb{F}_{q^2}$. The same problem is studied for the maximal curve corresponding to $sum_{t=1}^s y^{q/2^t}=x^{q+1}$ with $q=2^s$, over the finite field $mathbb{F}_{q^2}$.

Let $m>1$ be an integer and $Omega$ be an $m$-set‎. ‎The Hamming graph $H(n,m)$ has $Omega ^{n}$ as its vertex-set‎, ‎with two vertices are adjacent if and only if they differ in exactly one coordinate‎. ‎In this paper‎, ‎we provide a new proof on the automorphism group of the Hamming graph $H(n,m)$‎. ‎Although our result is not new (CE Praeger‎, ‎C Schneider‎, ‎Permutation groups and Cartesian decompositions‎, ‎Cambridge University Press‎, ‎2018)‎, ‎we believe that our proof is shorter and more elementary than the known proofs for determining the automorphism group of Hamming graph‎.