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Register a new userConnectivity concerning the last two subconstituents of a Q-polynomial distance-regular graph
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Let $Gamma$ be a $Q$-polynomial distance-regular graph of diameter $dgeq 3$. Fix a vertex $gamma$ of $Gamma$ and consider the subgraph induced on the union of the last two subconstituents of $Gamma$ with respect to $gamma$. We prove that this subgraph is connected.
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A connected graph $G$ is said to be $k$-connected if it has more than $k$ vertices and remains connected whenever fewer than $k$ vertices are deleted. In this paper, for a connected graph $G$ with sufficiently large order, we present a tight sufficient condition for $G$ with fixed minimum degree to be $k$-connected based on the $Q$-index. Our result can be viewed as a spectral counterpart of the corresponding Dirac type condition.
Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{alpha}(G)=alpha D(G)+(1-alpha)A(G) $$ for any real $alphain [0,1]$. The $A_{alpha}$-characteristic polynomial of $G$ is defined to be $$ det(xI_n-A_{alpha}(G))=sum_jc_{alpha j}(G)x^{n-j}, $$ where $det(*)$ denotes the determinant of $*$, and $I_n$ is the identity matrix of size $n$. The $A_{alpha}$-spectrum of $G$ consists of all roots of the $A_{alpha}$-characteristic polynomial of $G$. A graph $G$ is said to be determined by its $A_{alpha}$-spectrum if all graphs having the same $A_{alpha}$-spectrum as $G$ are isomorphic to $G$. In this paper, we first formulate the first four coefficients $c_{alpha 0}(G)$, $c_{alpha 1}(G)$, $c_{alpha 2}(G)$ and $c_{alpha 3}(G)$ of the $A_{alpha}$-characteristic polynomial of $G$. And then, we observe that $A_{alpha}$-spectra are much efficient for us to distinguish graphs, by enumerating the $A_{alpha}$-characteristic polynomials for all graphs on at most 10 vertices. To verify this observation, we characterize some graphs determined by their $A_{alpha}$-spectra.
In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius $rho$ equal to $3$ or $4$, and are $1/2^i$-th parts, for $iin{1,ldots,u}$ of binary (respectively, extended binary) Hamming codes of length $n=2^m-1$ (respectively, $2^m$), where $m=2u$. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter $D$ equal to $3$ or $4$ are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive.
Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schroder paths (resp. little Schroder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: begin{align*} r(n,0)&=sumlimits_{j=0}^{n-1}2^{j}r(n-1,j), r(n,k)&=r(n-1,k-1)+sumlimits_{j=k}^{n-1}2^{j-k}r(n-1,j),quad 1le kle n, s(n,0) &=sumlimits_{j=1}^{n-1}2cdot3^{j-1}s(n-1,j), s(n,k) &=s(n-1,k-1)+sumlimits_{j=k+1}^{n-1}2cdot3^{j-k-1}s(n-1,j),quad 1le kle n. end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,kge 0}$ and $[s(n,k)]_{n,kge 0}$, whose row sums produce the large and little Schroder numbers respectively, are two Riordan arrays of Bell type. Hence the above recurrences can also be deduced from their $A$- and $Z$-sequences characterizations. On the other hand, it is well-known that the large Schroder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run, whose distribution on separable permutations is shown to be given by $[r(n,k)]_{n,kge 0}$ as well.
We study a polynomial sequence $C_n(x|q)$ defined as a solution of a $q$-difference equation. This sequence, evaluated at $q$-integers, interpolates Carlitz-Riordans $q$-ballot numbers. In the basis given by some kind of $q$-binomial coefficients, the coefficients are again some $q$-ballot numbers. We obtain in a combinatorial way another curious recurrence relation for these polynomials.
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