Chemical Graph Theory - Part II
Org: Nino Bašić (University of Primorska, Slovenia) et Elizabeth Hartung (Massachusetts College of Liberal Arts, USA)
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PATRICK W. FOWLER, University of Sheffield, UK
The Chemical Significance of Graph Energy  [PDF]

Energy is a well-defined physical quantity with discrepant definitions in the mathematical chemistry of $\pi$ systems. In an extensive mathematical literature, graph energy, $E_{\rm G}(G)$, is the sum of absolute values of adjacency eigenvalues of graph $G$. This is a tractable but imperfect mimic of the physical Hückel energy, $E_\pi(G,N)$, a quantity that depends on both $G$, the molecular graph of the conjugated carbon framework, and the $\pi$ electron count, $N$. Discrepancies between $E_{\rm G}(G)$ and $E_\pi(G,N)$ can be arbitrarily large, but we reconcile the two definitions with a natural connection to the chemical concept of bond number.

IRENE SCIRIHA, University of Malta, Malta
The conductivity of the connected sum of root graphs with a common nullspace  [PDF]

Two connected root--graphs, $H_1$ and $H_2,$ with isomorphic subgraphs $H_1-z_1$ and $H_2-z_2$, are glued together to form their connected-sum $Z$. If their $\mu$--eigenspace is generated by vector ${\bf y}$ for some eigenvalue $\mu$ of their 0-1-adjacency matrix, then the $\mu$-multiplicity of $Z$ is shown to depend on the $\mu$--type of $z_1$ and $z_2$ in the root--graphs. A sufficient condition for the uniqueness of $H_1 (\simeq H_2),$ for a given $\bf y,$ when constructed from $H_1-z_1,$ is also established. The SSP model for ballistic conduction in a pi-molecule predicts that 5 out of the 11 feasible MEDs can be $Z$.

JELENA SEDLAR, University of Split, Croatia
Two types of indices and their extremal trees  [PDF]

We introduce the ordering of tree graphs so that the star $S_{n}$ is minimal and the path $P_{n}$ is maximal graph. Topological indices are of Wiener or anti-Wiener type, if they are increasing or decreasing functions with respect to the introduced ordering. If an index is of Wiener type $S_{n}$ is minimal and $P_{n}$ is maximal tree, for anti-Wiener type the reverse holds. We introduce a simple criterion to establish if a topological index is of Wiener or anti-Wiener type and apply our result to several generalizations of Wiener index.

RISTE ŠKREKOVSKI, University of Ljubljana, Slovenia
On 12-regular nut graphs  [PDF]

A nut graph is a simple graph whose adjacency matrix is singular with $1$-dimensional kernel and corresponding eigenvector with no zero elements. For each $d\in\{3,4,\dots,11\}$ are known all values $n$ for which there exists a $d$-regular nut graph of order $n$. In the talk, we consider all values $n$ for which there exists a $12$-regular nut graph of order $n$. (This is a joint work Nino Ba\v si\'c and Martin Knor.)

DRAGAN STEVANOVIĆ, Mathematical Institute of the Serbian Academy of Sciences and Arts, Serbia
On Hosoya's dormants and sprouts  [PDF]

Study of cospectral graphs is a traditional topic of spectral graph theory. Haruo Hosoya recently drew attention to a particular aspect of constructing cospectral graphs using coalescences: that cospectral graphs can be constructed by attaching multiple copies of the same rooted graph in different ways to subsets of vertices of an underlying graph. We address expectations and questions raised in Hosoya’s papers, and present an explicit formula for the characteristic polynomial of such multiple coalescences, establishing a necessary and sufficient condition for their cospectrality in the case when the attached rooted graph may be arbitrary.

(Joint work with Salem Al-Yakoob.)