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 \(\Gamma\).

In this paper, all connected graphs with the fourth largest signless-Laplacian eigenvalue less than two are determined.

The Lights Out game on a graph \(G\) is played as follows. Begin with a (not necessarily proper) coloring of \(V(G)\) with elements of \(\mathbb{Z}_2\). When a vertex is toggled, that vertex and all adjacent vertices change their colors from \(0\) to \(1\) or vice-versa. The game is won when all vertices have color \(0\). The winnability of this game is related to the existence of a parity dominating set.
We generalize this game to \(\mathbb{Z}_k\), \(k \geq 2\), and use this to define a generalization of parity dominating sets. We determine all paths, cycles, and complete bipartite graphs in which the game over \(\mathbb{Z}_k\) can be won regardless of the initial coloring, and we determine a constructive method for creating all caterpillar graphs in which the Lights Out game cannot always be won.

A total coloring of a simple graph \(G\) is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color.The minimum number of colors required for a proper total coloring of \(G\) is called the total chromatic number of \(G\) and denoted by \(\chi_t(G)\). The Total Coloring Conjecture (TCC) states that for every simple graph \(G\),\(\Delta(G) + 1 \leq \chi_t(G) \leq \Delta(G) + 2.\) \(G\) is called Type \(1\) (resp. Type \(2\)) if \(\chi_t(G) = \Delta(G) +1\) (resp. \(\chi_t(G) = \Delta(G) + 2\)). In this paper, we prove that the folded hypercubes \(FQ_n\), is of Type \(1\) when \(n \geq 4\).

Let \(H\) be a simple graph with \(n\) vertices and \(\mathcal{G} = \{G_1, G_2, \ldots, G_n\}\) be a sequence of \(n\) rooted graphs.
Following Godsil and McKay (Bull. Austral. Math. Soc. \(18 (1978) 21-28\)) defined the the rooted product \(H({G})\) of \(H\) by \({G}\) is defined by identifying the root of \(G_i\) with the \(i\)th vertex of \(H\).In this paper, we calculate the Wiener index of \(H({G})\), i.e., the sum of distances between all pairs of vertices, in terms of the Wiener indices of \(G_i\), \(i = 1, 2, \ldots, k\).As an application, we derive a recursive relation for computing the Wiener index of Generalized Bethe trees.

Let \(G\) be a connected graph with \(p\) vertices and \(q\) edges.A \(\gamma\)-labeling of \(G\) is a one-to-one function f from \(V(G)\) to \({0,1,…,q}\) that induces a labeling \(f’\) from \(V(G)\) to \({1,2,…,q}\) defined by \(f(e) = |f(u) – f(v)|\) for each edge \(e = uv\) of \(G\). The value of a \(\gamma\)-labeling \(f\) is defined to be the sum of the values of \(f’\) over all
edges. Also, the maximum value of a \(\gamma\)-labeling of \(G\) is defined as the maximum of the values among all \(\gamma\)-labelings of \(G,\) while the minimum value is the minimum of the values among all \(\gamma\)-labelings
of \(G\). In this paper, the maximum value and minimum value are determined for any complete bipartite graph.