A star edge coloring of a graph \(G\) is a proper edge coloring in which every path or cycle on four vertices contains at least three distinct colors. The star edge chromatic number of \(G\), denoted by \(\chi^\prime_{\mathrm{st}}(G)\), is the minimum number of colors required for such a coloring. In this paper, we study the star edge chromatic number of several graph classes derived from the Knödel graph \(W_{\Delta,n}\). In particular, we determine bounds and exact values for \(\chi^\prime_{\mathrm{st}}(G)\) for the Knödel graph \(W_{\Delta,n}\), its middle graph \(M\!\left(W_{\Delta,n}\right)\), and its Mycielskian \(\mu\!\left(W_{\Delta,n}\right)\). Furthermore, we provide explicit constructions of star edge colorings that attain these values.

Let \(G\) be a simple connected mixed graph, and let \(H(G)\) denote the Hermitian adjacency matrix of \(G\). The Hermitian permanental polynomial of \(G\) is defined as \(\pi(G; x) = \operatorname{per}(xI – H(G))\), where \(\operatorname{per}(\cdot)\) represents the permanent and \(I\) is the identity matrix. In this paper, we first derive fundamental properties of the Hermitian permanental polynomial for mixed graphs and establish explicit formulas relating its coefficients to those of the characteristic polynomial. We then analyze the root distribution of this polynomial, determining the number of zero roots for several special classes of mixed graphs. Finally, we characterize mixed graphs that remain cospectral under four‑way switching and prove that this operation preserves the permanental spectrum.

The Euler Sombor (\(EU\)) index of a graph \(G\) is defined as \[EU(\mathit{G})=\sum \limits_{{\mathit{u}}{\mathit{v}}\in E(\mathit{G})} \sqrt{deg_G^2(u)+deg_G^2(v)+deg_G(u)deg_G(v)},\] where \(deg_G(u)\) and \(deg_G(v)\) are the degrees of the vertices \(u\) and \(v\) in the graph \(G\), respectively. Biswaranja Khanra and Shibsankar Das [Euler Sombor index of trees, unicyclic and chemical graphs, MATCH Commun. Math. Comput. Chem., 94 (2025) 525–548] posed an open problem to determine the extremal values and extremal graphs of the Euler Sombor index in the class of all connected graphs with a given domination number. In this paper, we solve this open problem for trees with a given domination number. Furthermore, we determine an upper bound for the Euler Sombor index of trees with a given independence number. We also characterize the corresponding extremal trees. Additionally, we propose a set of open problems for future research.

The exponential Randić index of a graph \(G\), denoted by \(ER(G)\), is defined as \(\sum\limits_{uv\in E(G)}e^{\frac{1}{d(u)d(v)}}\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). The line graph \(L(G)\) of a graph \(G\) is a graph in which each vertex represents an edge of \(G\), and two vertices are adjacent in \(L(G)\) if and only if their corresponding edges in \(G\) are incident to a common vertex. In this paper, we proved that for any tree \(T\) of order \(n\ge3\), \(ER(L(T))>\frac{n}{4}e^{\frac{1}{2}}\).

A \(\{P_5^{\alpha}, S_5^{\beta}, Y_5^{\gamma}\}\)-decomposition of a graph is a partition of its edge set into \(\alpha\) copies of \(P_5\), \(\beta\) copies of \(S_5\), and \(\gamma\) copies of \(Y_5\), where \(P_5\), \(S_5\), and \(Y_5\) denote the three non-isomorphic trees of order five. In this paper, we study the existence of a \(\{P_5^{\alpha}, S_5^{\beta}, Y_5^{\gamma}\}\)-decomposition of the complete bipartite graph \(K_{m,n}\) for \(m\geq 4\) and \(n\geq 2\), and of the complete graph \(K_n\) for \(n\geq 8\). In fact, we establish necessary and sufficient conditions for the existence of such decompositions in \(K_{m,n}\) and \(K_n\).

Given a connected graph \(G\) and a configuration \(D\) of pebbles on \(V(G)\), a pebble move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A monophonic path is a longest chordless path between two non-adjacent vertices \(u\) and \(v\). The line segment that connects two vertices on a curve is known as a chord. The monophonic distance between \(u\) and \(v\) is the number of vertices in the longest \(u\)–\(v\) monophonic path, denoted by \(d_{\mu}(u,v)\) in \(G\). The monophonic pebbling number (MPN) of \(G\) is the least number of pebbles needed to guarantee that, from any distribution of pebbles on a graph \(G\), one pebble can be moved to any specified vertex using monophonic paths through pebbling moves. The monophonic \(t\)-pebbling number (MtPN) of \(G\) is the least number of pebbles needed to guarantee that, from any distribution of pebbles, \(t\) pebbles can be moved to any specified vertex using monophonic paths. In this article, we determine the \(MPN\) and \(MtPN\) of Dutch windmill graphs, square of cycles, tadpole graphs, lollipop graphs, double star path graphs, and fuse graphs, and we also discuss their \(t\)-pebbling versions.