In this paper, we generalize to the class of signed graphs the well known result that every numbered graph can be embedded as an induced subgraph in a gracefully numbered graph.

There are \(267\) nonisomorphic groups of order \(64\). It was known that \(259\) of these groups admit \((64, 28, 12)\) difference sets and the other eight groups do not admit \((64, 28, 12)\) difference sets. Despite this result, no research investigates the problem of finding all \((64, 28, 12)\) difference sets in a certain group of order \(64\).In this paper, we find all \((64, 28, 12)\) difference sets in \(111\) groups of order \(64. 106\) of these groups are nonabelian. The other five are \(\mathbb{Z}_{16} \times \mathbb{Z}_4\), \(\mathbb{Z}_{16} \times \mathbb{Z}_2^2\), \(\mathbb{Z}_8 \times \mathbb{Z}_8\), \(\mathbb{Z}_8 \times \mathbb{Z}_4 \times \mathbb{Z}_2\), and \(\mathbb{Z}_8 \times \mathbb{Z}_2^3\).In these \(111\) groups, we obtain \(74,922\) non-equivalent \((64, 28, 12)\) difference sets. These difference sets provide at least \(105\) nonisomorphic symmetric \((64, 28, 12)\) designs. Most of our work was done using programs with the software \(GAP\).

In this paper, we obtain some generating functions for the generalized Zernike or disk polynomials \(P_{m,n}^\alpha (z,z^*)\) which are investigated by Wiinsche [13]. We derive various families of bilinear and bilateral generating functions. Furthermore, some special cases of the results presented in this study are indicated. Also, it is possible to obtain multilinear and multilateral generating functions for the polynomials \(P_{m,n}^\alpha (z,z^*)\).

A \((k,t)\)-list assignment \(L\) of a graph \(G\) is a list of \(k\) colors available at each vertex \(v\) in \(G\) such that \(|\bigcup_{v\in V(G)}L(v)| = t\). A proper coloring \(c\) such that \(c(v) \in L(v)\) for each \(v \in V(G)\) is said to be an \(L\)-coloring. We say that a graph \(G\) is \(L\)-colorable if \(G\) has an \(L\)-coloring. A graph \(G\) is \((k,t)\)-choosable if \(G\) is \(L\)-colorable for every \((k,t)\)-list assignment \(L\).
Let \(G\) be a graph with \(n\) vertices and \(G\) does not contain \(C_5\) or \(K_{k-2}\) and \(K_{k+1}\). We prove that \(G\) is \((k, kn – k^2 – 2k)\)-choosable for \(k \geq 3\).\(G\) is not \((k, kn – k^2 – 2k)\)-choosable for \(k = 2\).This result solves a conjecture posed by Chareonpanitseri, Punnim, and Uiyyasathian [W. Chareonpan-itseri, N. Punnim, C. Uiyyasathian, On \((k,t)\)-choosability of Graphs: Ars Combinatoria., \(99, (2011) 321-333]\).

We call a graph \(G\) a \({generalized \;split \; graph}\) if there exists a core \(K\) of \(G\) such that \(V(G) \setminus V(K)\) is an independent set of \(G\).Let \(G\) be a generalized split graph with a partition \(V(G) = K \cup S\), where \(K\) is a core of \(G\) and \(S\) is an independent set. We prove that \(G\) is end-regular if and only if for any \(a, b \in S\), \(\phi \in \text{Aut}(K)\), the inclusion \(\phi(N(a)) \subsetneqq N(b)\) does not hold.
\(G\) is end-orthodox if and only if \(G\) is end-regular and for any \(a, b \in S\), \(N(a) \neq N(b)\).

In this paper we generalize the Fibonacci numbers and the Lucas numbers with respect to \(n\), respectively \(n+1\) parameters. Using these definitions we count special subfamilies of the set of \(n\) integers. Next we give the graph interpretations of these numbers with respect to the number of \(P_k\),-matchings in special graphs and we apply it for proving some identity and also for counting other subfamilies of the set of n integers.

The Wiener-Hosoya index was firstly introduced by M. Randié¢ in \(2004\). For any tree \(T\), the Wiener-Hosoya index is defined as

\[WH(T)= \sum\limits_{e\in E(T)} (h(e) + h[e])\]

where \(e = uv\) is an arbitrary edge of \(T\), and \(h(e)\) is the product of the numbers of the vertices in each component of \(T – e\), and \(h[e]\) is the product of the numbers of the vertices in each component of \(T- \{u,v\}\). We shall investigate the Wiener-Hosoya index of trees with diameter not larger than \(4\), and characterize the extremal graphs in this paper.

Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and
successive derivatives of binomial type sequences. We give some relations between Bell polynomials and binomial type sequences in
first part, and, we generalize the results obtained in \([4]\) in second part.

Fouquet and Jolivet conjectured that if \(G\) is a \(k\)-connected \(n\)-vertex graph with independence number \(\alpha \geq k \geq 2\), then \(G\) has circumference at least \( \frac{k(n+\alpha-k)}{\alpha} \). This conjecture was recently proved by \(O\), West, and Wu.
In this note, we consider the set of \(k\)-connected \(n\)-vertex graphs with independence number \(\alpha > k \geq 2\) and circumference exactly \( \frac{k(n+\alpha-k)}{\alpha} \). We show that all of these graphs have a similar structure.

Let \(\Gamma\) be the rank three \(M_{24}\) maximal \(2\)-local geometry. For the two conjugacy types of involution in \(M_{24}\), we describe the fixed point sets of chambers in \(\Gamma\).