n this paper‎, ‎we investigate suborbital graphs formed by $Nbig(Gamma_0(N)big)$-invariant equivalence relation induced on $hat{mathbb{Q}}$‎. ‎Conditions for being an edge are obtained as a main tool‎, ‎then necessary and sufficient conditions for the suborbital graphs to contain a circuit are investigated‎.

In this paper‎, ‎we evaluate determinants of some families of Toeplitz--Hessenberg matrices having tribonacci number entries‎. ‎These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers‎. ‎In particular‎, ‎we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz--Hessenberg determinants‎. ‎We then obtain‎, ‎by combinatorial arguments‎, ‎extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying a recurrence of the form $T_n^{(r)}=T_{n-1}^{(r)}+T_{n-2}^{(r)}+T_{n-r}^{(r)}$ for $n geq r$‎, ‎with the appropriate initial conditions‎, ‎where $r geq 3$ is arbitrary‎.

We prove that the graph obtained from the non-nilpotent graph of a finite group by deleting the isolated vertices is connected with diameter at most 3‎. ‎This bound is the best possible‎.

‎We define a refinement of the notion of Leray simplicial complexes and study its properties‎. ‎Moreover‎, ‎we translate some of our results to the language of commutative algebra‎.

In this paper we classify distance-regular graphs‎, ‎including strongly regular graphs‎, ‎admitting a transitive action of the linear groups $L(3,2)$‎, ‎$L(3,3)$‎, ‎$L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15‎. ‎We give details about constructed graphs‎. ‎In addition‎, ‎we construct self-orthogonal codes from distance-regular graphs obtained in this paper‎.

‎‎The generalized Zagreb index is an extension of both ordinary and‎ ‎variable Zagreb indices‎. ‎In this paper‎, ‎we present exact formulae‎ ‎for the values of the generalized Zagreb index for product graphs‎. ‎Results are applied to some graphs of general and chemical‎ ‎interest such as nanotubes and nanotori‎.

We consider some combinatorics of elliptic root systems of type $A_1$. In particular, with respect to a fixed reflectable base, we give a precise description of the positive roots in terms of a ``positivity'' theorem. Also the set of reduced words of the corresponding Weyl group is precisely described. These then lead to a new characterization of the core of the corresponding Lie algebra, namely we show that the core is generated by positive root spaces.

‎‎‎For a simple connected graph $G$ of order $n$ and size $m$‎, ‎the Laplacian energy of $G$ is defined as‎ ‎$LE(G)=sum_{i=1}^n|mu_i-frac{2m}{n}|$ where $mu_1‎, ‎mu_2,ldots‎,‎‎mu_{n-1}‎, ‎mu_{n}$‎ ‎are the Laplacian eigenvalues of $G$ satisfying $mu_1ge mu_2gecdots ge mu_{n-1}>‎ ‎mu_{n}=0$‎. ‎In this note‎, ‎some new lower bounds on the graph invariant $LE(G)$ are derived‎. ‎The obtained results are compared with some already known lower bounds of $LE(G)$‎.

A set $D$ of vertices of graph $G$ is called $double$ $dominating$ $set$ if for any vertex $v$, $|N[v]\cap D|\geq 2$. The minimum cardinality of $double$ $domination$ of $G$ is denoted by $\gamma_d(G)$. The minimum number of edges $E'$ such that $\gamma_d(G\setminus E)>\gamma_d(G)$ is called the double bondage number of $G$ and is denoted by $b_d(G)$. This paper determines that $b_d(G\vee H)$ and exact values of $b(P_n\times P_2)$, and generalized corona product of graphs.

Let $G$ be a graph‎. ‎A function $f‎ : ‎V (G) longrightarrow {0,1}$‎, ‎satisfying‎ ‎the condition that every vertex $u$ with $f(u) = 0$ is adjacent with at‎ ‎least one vertex $v$ such that $f(v) = 1$‎, ‎is called a dominating function $(DF)$‎. ‎The weight of $f$ is defined as $wet(f)=Sigma_{v in V(G)} f(v)$‎. ‎The minimum weight of a dominating function of $G$‎ ‎is denoted by‎ ‎$gamma (G)$‎, ‎and is called the domination number of $G$‎. ‎A dominating‎ ‎function $f$ is called a global dominating function $(GDF)$ if $f$ is‎ ‎also a $DF$ of $overline{G}$‎. ‎The minimum weight of a global dominating function is denoted by‎ ‎$gamma_{g}(G)$ and is called global domination number of $G$‎. ‎In this paper we introduce a generalization of global dominating function‎. ‎Suppose $G$ is a graph and $sgeq 2$ and $K_n$ is the complete graph on $V(G)$‎. ‎A function $ f:V(G)longrightarrow { 0,1} $ on $G$ is $s$-dominating function $(s-DF)$‎, ‎if there exists some factorization ${G_1,ldots,G_s }$ of $K_n$‎, ‎such that $G_1=G$ and $f$ is dominating function of each $G_i$‎.