An \(f\)-coloring of a graph \(G\) is an edge-coloring of \(G\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. A multi-wheel graph is a graph obtained from \(s\) cycles \(C_{n_1}, C_{n_2}, \ldots, C_{n_s}\) (\(s \geq 1\)) by adding a new vertex, say \(w\), and edges joining \(w\) to all the vertices of the \(s\) cycles. In this article, we solve a conjecture posed by Yu et al. in 2006 and prove that it is not always true. Furthermore, the classification problem of multi-wheel graphs on \(f\)-colorings is solved completely.

For a connected graph \(G = (V, E)\) of order at least two, a chord of a path \(P\) is an edge joining two non-adjacent vertices of \(P\). A path \(P\) is called a monophonic path if it is a chordless path. A longest \(x\)-\(y\) monophonic path is called an \(x\)-\(y\) detour monophonic path. A set \(S\) of vertices of \(G\) is a detour monophonic set of \(G\) if each vertex \(v\) of \(G\) lies on an \(x\)-\(y\) detour monophonic path for some \(x\) and \(y\) in \(S\). The minimum cardinality of a detour monophonic set of \(G\) is the detour monophonic number of \(G\) and is denoted by \(dm(G)\). For any two vertices \(u\) and \(v\) in \(G\), the monophonic distance \(dm(u,v)\) from \(u\) to \(v\) is defined as the length of a \(u\)-\(v\) detour monophonic path in \(G\). The monophonic eccentricity \(em(v)\) of a vertex \(v\) in \(G\) is the maximum monophonic distance from \(v\) to a vertex of \(G\). The monophonic radius \(rad_{m}(G)\) of \(G\) is the minimum monophonic eccentricity among the vertices of \(G\), while the monophonic diameter \(diam_{m}(G)\) of \(G\) is the maximum monophonic eccentricity among the vertices of \(G\). It is shown that for positive integers \(r\), \(d\), and \(n \geq 4\) with \(r < d\), there exists a connected graph \(G\) with \(rad_{m}(G) = r\), \(diam_{m}(G) = d\), and \(dm(G) = n\). Also, if \(p\), \(d\), and \(n\) are integers with \(2 \leq n \leq p-d+4\) and \(d \geq 3\), there is a connected graph \(G\) of order \(p\), monophonic diameter \(d\), and detour monophonic number \(n\). Further, we study how the detour monophonic number of a graph is affected by adding some pendant edges to the graph.

In this paper we introduce a new kind of two-parameters generalization of Pell numbers. We give two distinct graph interpretations and prove some identities for these numbers. Moreover we define matrix generators and derive the generalized Cassini formula for the introduced numbers.

A graph is said to be a neighbourly irregular graph (or simply an NI graph) if no two adjacent vertices have the same degree. In this paper, we introduce the neighbourly regular strength of a graph. Let \(G\) be a simple graph of order \(n\). Let \(NI(G)\) denote the set of all NI graphs in which \(G\) is an induced subgraph. The neighbourly regular strength of \(G\) is denoted by \(NRS(G)\) and is defined as the minimum \(k\) for which there is an NI graph \(NI(G)\) of order \(n+k\) in \(NI(G)\). We prove that the \(NRS(G)\) is at most \(n-1\), with possible equality only if \(G\) is complete. In addition, we determine the \(NRS\) for some well-known graphs.